Карпатські математичні публікації T2 N1

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i

i

ii

.2, 1

2010

i

.., .. i i,

2-i i i . . . . . . . . . . . . . . . . . . . . . . . 4

.. i i

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

i .. i ii . . . . . . 24

i .. ii i ii ii i i i-

i i . . . . . . . . . . . . . . . . . . . . . . 35

.., i .. i i

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

.., .. ii i 72

.I., .., .. ii i

i i .. i i ii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

.., i .. i i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

.., .., i ., .. i

i i- i i-

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

i .., .. i ii i i

i i . . . . . . . . . . . . . . . . . . . . . . . . . . 109

I. ., i . . ii i

i . . . . . . 119

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Carpathian

mathematical

publications

Scientific journal

V.2, 1

2010

Contents

Voloshyn H.A., Maslyuchenko V.K. On approximation of the separately continuous

functions 2-periodical in relation to the second variable . . . . . . . . . . . . . . 4

Voloshyna T.V. An analogue of Bernsides lemma for finite inverse symmetric semi-

group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Gavrylkiv V.M. On representation of semigroups of inclusion hyperspaces . . . . . . . 24

Hryniv R.O. Analyticity and uniform stability in the inverse spectral problem for

impedance SturmLiouville operators . . . . . . . . . . . . . . . . . . . . . . . . . 35

Zagorodnyuk A.V., Kravtsiv V.V. Symmetric polynomials on the product of Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Zatorsky R.A., Semenchuk A.V. Periodic recurrent fractions of third degree . . . . . . 72

Kopach M.I., Obshta A.F., Shuvar B.A. On applications of iteration algorithms and

Skorobagatkos branching fractions to approximation of roots of polynomials in

Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Lopushansky O.V., Oleksienko M.V. A Poisson type formula for Hardy classes on

Heisenbergs group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Prykarpatsky A.K., Artemovich O.D., Popowicz Z., Pavlov M.V. Riemann type

algebraic structures and their differential-algebraic integrability analysis . . . . . . 96

Skaskiv O.B., Kuryliak A.O. Direct analogues of Wimans inequality for analytic

functions in the unit disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chuchman I. Ya., Gutik O. V. Topological monoids of almost monotone injective

co-finite partial selfmaps of the set of positive integers . . . . . . . . . . . . . . . . 119

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.2, 1

2010

.., .. ,

2- . . . . . . . . . . . . . . 4

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

.. . . . 24

..

. . . . . . . . . . . . . . . . 35

.., ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

.., .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

.., .., ..

..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

.., ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

.., .., ., ..

- -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

.., ..

. . . . . . . . . . . . . . . . . . . . . . . . . 109

. ., . . -

. 119

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i i Carpathian Mathematical

ii. .2, 1 Publications. V.2, .1

517.51

.., ..

I I,

2-I I I

.., .. i i, 2-

i i i // i i ii. 2010. .2,

1. C. 414.

i i , -

i X i i i i f : X R R,

2-i i i, i ii

i fn : X R R, i fx

n= fn(x, ) : R R i

i i fxn fx R x X.

i ii

ii i i [8, c. 98], : i i f : [0, 1]2 R i i-

i i fn : [0, 1]2 R, i i i ii

i , ii ii fxn = fn(x, ) x [0, 1] i-

i i fx = f(x, ) ii [0, 1]. i [1]

i Bn : C[0, 1] C[0, 1], i i g C[0, 1]

i

(Bng)(y) =n

k=0

Ckngk

n

yk(1 y)nk,

0 y 1, , i i X i i f : X [0, 1] R i

fn(x, y) = (Bnfx)(y),

x X, y [0, 1] i ii i i,

fxn fx [0, 1] x X. hn h Y ,

, ii ii hn h, hn h = supxY

|hn(y) h(y)| 0 n .

2000 Mathematics Subject Classification: 54C30, 65D15.i i : i i i, , .

c .., .., 2010

8/6/2019 T2 N1

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i i 5

ii (. [2]) i i i -

i i. i , -

ii

i [8, c. 398]. i -

, i X i i i

i f : XR R, 2-i i i, i i-i i fn : XR R, if

xn : R R

i i, i fxn fx R x X. ,

i

f : X R R, i 2-i i i, , -

i i ii ii, i i

ii.

i [3].

1 i ii

i C(Y) i i g : Y R, i-

i Y, i Tp ii,

i

qy(g) = |g(y)|, y Y.

i (C(Y), Tp) Cp(Y).

Y Cu(Y) i i

(C(Y), ) ii

g = maxyY

|g(y)|,

Tu i Cu(Y), i ii ii

C(Y).

A : C(Y) C(Y) pu-, i -

i A : Cp(Y) Cu(Y). pp-, uu- i up-

i.

C2 i i 2-i i

g : R R. i i

g = maxyR

|g(y)| = max0y2

|g(y)|,

C2 i ii ii Tu. i C2 -

i i Tp ii. iTp i Tu C2 iii

i i A : C2 C2. , A : C2 C2

pu-, i i A : (C2, Tp) (C2, Tu).

S = {z C : |z| = 1} i, i -

C. i, S i, i

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6 .., ..

Cu(S) i Cp(S). i i i h C(S) i g = U h C2,

g(y) = h(eiy)

y R. i, i U : C(S) C2 -

ii, ii i Cu(S) i (C2, ) i-

ii i Cp(S) i (C2, Tp).

ii C2 i C(S).

ii C0[0, 2] C[0, 2], -

i g C[0, 2], g(0) = g(2). i R : C2 C0[0, 2],

Rg = g|[0,2] ii, ii,

C2 i C0[0, 2] i ii . , i C2 -

i i ii C0[0, 2] C[0, 2].

i f : X Y Z i (x, y) X Y fx(y) = fy(x) =

f(x, y) i (x) = fx. i : X ZY -

f i f i. i i X Y C(X Y)

CC(X Y)

i (i)

i f : X Y R.

i (., ,

[7, 2.1.2]).

1. X iY ii , f : XY R i i : X RY

i f i. i:

) f CC(X Y) (X) C(Y) i i : X Cp(X) ;

) Y i, f C(X Y) (X) C(Y) i i : X Cu(X) .

i iii f i , 1

CC(X Y) C(X, Cp(Y)) i i-

: X Cp(Y), i p-, C(X Y)

C(X, Cu(Y)) i i : X Cu(Y), i -

u-.

i X CC2(X R) i i

i i f : X R R, i 2-i i

i. ii U : C(S) C2 i CC(X S), i CC2(XR) C(XR) C(X S), i

i i 1.

2. X i i, f : XR R i i : X RR i f i. i:

) f CC2(X R) (X) C2 i : X C2 p-;

) f CC2(X R) C(X R) (X) C2 i : X C2 u-.

i T i i ii g : R R,

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i i 7

i

g(y) = a0 +n

k=1

(ak cos ky + bk sin ky).

, T ii C2.

CT(X R) i f : X R R, i

fy : X R i y R, ifx : R R ii x X. i, CT(X R) ii

CC2(X R).

2 i ii i

ii i An : C2 T T,

Ang g R i g C2.

= p, u i = p, u. ii i An : C2 T

-, i An -. A : C2 C2

A : CC2(X R) CC2(X R), i i f CC2(X R) iii i f = Af, X R

f(x, y) = (Afx)(y).

(

x) =

fx

i (x) = fx x X, f CC2(X R) i f = Af.i, i = A . i i , i u-, p- i A pu- u- i A

uu-. i, 2, .

3. ) A : C2 C2 pu- , f CC2(X R) i

f =

Af. i

f CC2(X R) C(X R).

) A : C2 C2 uu- , f CC2(X R) C(X R) if = Af. i f CC2(X R) C(X R). i .

4. (An)n=1 ii i An : C2 C2, f CC2(X R)i fn = Anf. i:

) (An)n=1 pu- ii i T, fn

CT(X R) C(X R) n i fxn fx R x X;

) f C(XR) i(An)n=1 uu- ii i T,

fn CT(X R) C(X R) n i fxn f

x R x X.

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8 .., ..

. i An(C2) T, fn CT(XR). i i

fn 3. i,

fxn = An((x)) (x) = fx R

x X, ii i An .

3

i i ii ii:

Dn(t) =sin(2n + 1) t

2

2sin t2

=1

2+

nk=1

cos kt,

i :

Kn(t) = 2n + 1

sin(n + 1) t22sin t

2

2 = 1n + 1

nk=0

Dk(t),

i . ii Dn n- -

(Sng)(x) =a0

2+

nk=1

(ak cos kx + bk sin kx)

i g C2 [8, c. 381]:

(Sng)(x) = 1

0

(g(x + t) + g(x t))Dn(t)dt,

i i [7, c.751]:

(Fng)(x) =1

n + 1

nk=0

(Skg)(x) =1

0

(g(x + t) + g(x t))Kn(t)dt.

i [7, c. 526; c. 752], Sng g i-

i g C2, i Fng

g R

i g C2. Fn . i Fng T n i

Fng g R i g C2, (Fn)n=1 ii

i i T i ii.

i i g C2:

(Fng)(x) =1

g(x + t)Kn(t)dt =1

g(y)Kn(y x)dy.

, ii i 2-i.

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i i 9

i , , Kn i i:

1) Kn(0) =n+12

;

2) Kn(t) 0 i Kn(t) = Kn(t) i t R;

3)1

K

n(t)dt = 1.

i, Fn : C2 C2 uu-.

i, Fn ii i i g C2 i i x R

|Fng(x)| 1

|g(x + t)|Kn(t)dt g

Kn(t)dt = g,

i ii Fng g, i Fn i (C2, ). i Tp Tu, Fn i up-

. , .

5. ) Fn : C2 T uu- i-i i T i ii.

) i i X i i 2-

i i i i f : X R R i

fn(x, y) = (Fnfx)(y)

, fxn T - n N i x X fxn f

x R

i x X.

i, Fn pu-.

. K C2, K(t) 0 R, K(0) > 0 i t0 R. i i

(g) =

g(t)K(t t0)dt

i (C2, Tp).

. a [, ), t0 = a + 2m i m. = K(0)

2. , 0 < < K(0). i i K i,

i > 0, a + < i K(t) , i a t a + . n

bn = a+n

i i gn C2, i ii [, ]

(, 0), (a, 0), (a + bna2

, n), (bn, 0) (, 0). ,

gn(t) 0 t R, bn a n . , i K(t t0) = K(t a)

t R,

(gn) =

gn(t)K(t t0)dt =

gn(t)K(t a)dt

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10 .., ..

=

bna

gn(t)K(t a)dt

bna

gn(t)dt = n(bn a)

2=

2> 0

n. , (gn) 0. i , i

i Tp ii.

i .

6. Fn pp-, , i pu-.

. i

(Fng)(x) =1

g(y)Kn(y x)dy,

i K = 1

Kn C2 i K(0) =n+12

> 0, i ,

x R i

x(g) = (Fng)(x) =

g(y)Kn(y x)dy

p- C2. i Fn pp-, pp-

i i p-i i ii x,

.

4

i ii [0, 2] n + 1 i xk =2kn+1

. 0 a0 x1i ak = a0 + kd k = 1, . . . , n, d =

2n+1

. a0, a1, . . . , an ii ii

ii [0, 2], ak+1 ak = d i xk ak xk+1 k = 0, 1, . . . , n.

i g C2, n i x R

(Jng)(x) =2

n + 1

nk=0

g(ak)Kn(x ak),

Kn . i Kn i, i Jng

i. , i Jn : C2 T,i, , ii. i Jn , i

Jng i ig. i, Jng g R i

g C2 [4, c.36]. i i

i i

(Jng)(x) =1

g(y)Kn(y x)dn+1(y),

n+1(y) = kd, ak < y ak+1, i d =2

n+1

.

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i i 11

7. ) Jn : C2 T pu- i-i T.

) i i X i f CC2(X R)

i

fn(x, y) = (Jnfx)(y)

i CT(X R), fxn fx R x X.

. ) i y1, . . . , yn R i i 1, . . . , n C2

(Ag)(y) =n

k=1

g(yk)k(y)

i Ag C2, i A : C2 C2, , -

, ii. g C2

Ag maxk=1,...,n

|g(yk)| = maxk=1,...,n

qyk(g),

=n

k=1

k. i , A pu- [5, .12].

i, Jn . ,

pu-i. i Jng g R g C2 i imJn T,

) .

) ) i 4.

5 i

ii pu- ii i Gn : C2

T T, i Bn : C[0, 1] C[0, 1] (. .1).

[8, c. 398].

P[a, b] i i ii ii [a, b]. Bni C[0, 1] P[0, 1] i pu-. Bnf f [0, 1]

f C[0, 1].

ii i : [0, 1] [1, 1], (x) = 2x 1, -

i U : C[1, 1] C[0, 1], U f = f . i y = (x) -

i x = 1(y) = 12

(y + 1), ii, U

U1 : C[1, 1] C[0, 1], i U1g = g 1.

Cn = U1BnU.

i i i ii, U1(P[0, 1]) P[1, 1], , im Cn P[1, 1].

U i U1 pp- i uu-, , Cn pu-

i , i i C[1, 1] P[1, 1]. i , Cnf U1U f = f

[1, 1] f C[1, 1], U1 uu-.

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12 .., ..

C+2

C2

ii i () i C2. -

(t) = arccos t. i : [1, 1] [0, ] i

i 1(x) = cos x. V f = f . V : C[0, ] C[1, 1]

. V1g = g 1. V i V1 pp-

i uu-i. i , g P[1, 1] i f = V1g, f(x) = g(cos x)

0 x . i , f - ii [0, ] , T+[0, ],

i i T C+2.

R : C2 C[0, ], Rf = f|[0,] i P : C[0, ] C2,

iii i i g C[0, ] i f C+2,

f|[0,] = g. R i P pp- i uu-, P Rf = f,

f C+2.

Dn = P V1CnV R.

Dn : C2 C2 pu-, V R : C2 C[1, 1] pp-

, Cn : C[1, 1] C[1, 1] pu- i P V1

:C[1, 1] C2 uu-.

im Dn P V1(im Cn) P V

1(P[1, 1]) P(T+[0, ]) T C+2

i f C+2

Dnf P V1V Rf = P Rf = f R.

K : C2 C+2 i L : C2 C

2,

(Kf)(x) =f(x) + f(x)

2i (Lf)(x) =

f(x) f(x)

2 i f C2 i x R. i, K + L = I, I .

K i L pp- i uu-. M : C2 C2

, N : C2 C2 ,

(M f)(x) = f(x)sin x i (N f)(x) = f(x)cos x

f C2 i x R. M N pp- i uu-i. ,

M(T) T i N(T) T, M(C2) C+2.

En = M2DnK + M DnM L.

i, En : C2 C2 pu- , im En T.

f C2 , Kf C+2 i M Lf C

+2.

Enf = M2DnKf + M DnM LfM

2Kf + MMLf = M2(Kf + Lf) = M2f R.

W : C2 C2, W f = f , (x) = x +2

, i

W1, W1g = g 1, 1(x) = x 2

. -

i, W i W1 pp- i uu-, W(T) T i W1(T)

T.

Fn = W1EnW.

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i i 13

, im Fn W1(imEn) W

1(T) T. Fn pu- i

f C2

Fnf = W1EnW fW

1M2W f R,

W1M2W f(x) = W1M2f(x + 2

) = W1[f(x + 2

)sin2 x]

= f(x

2+

2)sin2(x

2) = f(x)cos2 x = N2f(x).

,

Fnf N2f R.

i

Gn = En + Fn.

Gn pu-, En i Fn . i, im Gn T,

im En T im Fn T.i, f C2

Gnf = Enf + FnfM2f + N2f = (M2 + N2)f = f R,

M2 + N2 = I, sin2 x + cos2 x = 1 i x R.

, .

8. Gn : C2 T pu- ii T.

i , Gn - Bn. :

Gn = En + Fn

= M2DnK + M DnM L + W1M2DnKW + W

1M DnM LW

= M2P V1CnV RK + M P V1CnV R M L + W

1M2P V1CnV RK W

+W1M P V1CnV RMLW = M2P V1U1BnU V R K

+M P V1U1BnU V R M L + W1M2P V1U1BnU V R K W

+W1

M P V

1

U

1

BnUV RMLW.

En = M2P V1U1BnU V R K + M P V

1U1BnU V R M L

i

Gn = En + W1EnW.

, [7, c.698] i

i C2 i,

ii- ii ii i ,

i pu- ii i An : C2 T.

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14 .., ..

i

1. .., .. i i i i // . i.

i. -. . 2007. . 336-337. . 52-59.

2. .., .., .. i i

i // . . i i i i i.

i. I-i. 2010. . 2-3.

3. .., .. i i, 2-i i-

i // . . i i i i

i. i. I-i. 2010. . 27-28.

4. . . .2. M.: , 1965. 538 .

5. .. iii i . ii: , 2002. 72 .

6. .. i i i i // . . i.-. .

ii. 1999. 345 .

7. .. . .III. .-.: . -

.-. -, 1949. 783 .

8. .. . T 2. C.---: ,

2005. 464 .

i i i ii i ,

ii,

i 7.04.2010

Voloshyn H.A., Maslyuchenko V.K. On approximation of the separately continuous functions

2-periodical in relation to the second variable, Carpathian Mathematical Publications, 2, 1

(2010), 414.

Using Jacksons and Bernsteins operators we prove that for every topological space X and

an arbitrary separately continuous function f : X R R, 2-periodical in relation to the

second variable, there exists such sequence of jointly continuous functions fn : X R R

such that functions fxn

= fn(x, ) : R R are trigonometric polynomials and fx

n fx on R

for every x X.

.., .. , 2-

// -

. 2010. .2, 1. C. 414.

,

X f :

X R R, 2- ,

fn : X R R,

fxn

= fn(x, ) : R R fxn fx R

x X.

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512.53

..

I I

I

.. i i i-

// i i ii. 2010. .2, 1. C. 1523.

i

i i i.

S i i. S: ab aa1 = ab1 a1a = a1b. i ii

. H H S H := {h S : H h}. H = H, H .

i i i X i i i X i - IS(X) ISn, |X| = n. i IS(X) - i X. i i iS i i i i IS(X) i i X. i i i i- i i , iii .

i ii H i i S - i H :=

(s, t) S S

st1 H. ii i (Hs), ss1 H, i i ii H. i X i H i(S) : x X i s S xH(s) = (xs). : SIS(X) i S

2000 Mathematics Subject Classification: 20M18, 20M20, 20M30.i i : , i i.

c .., 2010

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16 ..

i iiH. i [1] i, - i i S i i i i i ii.

N = {1, 2, . . . n}, ISn i i i N,S(M) i M. ISn dom i ran -

ii i . i i i , - ii . i i rank i def ii.

i i ii i i .

1 ([1]). i M N i i G S(M) iiH = G ISM i ii ISn. i ,

i ii ISn .

i ii H = G ISM |M| = k,|G| = m, [S(M) : G] = k!

m= r. ii i i ISn -

i ii H [ISn : H].

2 ([1]). i iISn i- ii H = G ISM (Hg), g S(N), i i

ii Ag S(N) i A = GS(M), iii

i i ii i A .

[ISn : H] = S(N) : G S(M) =n!

m (n k)!= Ck

n

[S(M) : G] = Ckn

r.

i i ii iii H.

, i , [3]. : ISn IS(X) i i ISn i X

i i ii H. -i 1 = {(x, y) ISn ISn|(x) = (y)} i i iISn. ii i ii i i i iii H [2].

3 ([1]). i iISn i i i ii H = G ISM i i

i i, |M| = 1. i

ISn i ISn i i

{1, 2, . . . , n}.

|(ISn| = |ISn| =ni=0

(Cin)2 i!.

1 ([2]). i ii H = G ISM, |M| = k, Ik = { ISn|rank < k} ii i

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i i i 17

1. t Ik (t) i -

i i i i ii

H = G ISM .

. k = n, M = N i H = G S(N). 1, i i ISn i i. (ISn) i .

1

() ii i .

2. i i ISn ii

ISn

() +1

n

ISn

def = |ISn| . (1)

. , |ISn| =n

k=0

Ckn2k!. 1n

ISn

def .

i i. k in k. ii i k ii ISn i

Ckn2 k!.

1

n

ISn

def =1

n

nk=0

ISn

rank=k

def =1

n

nk=0

(n k)

Ckn2

k!.

ISn

() i i. i

k Ckn . i ISn i k i i i ii. i i i L N i k i

ISn

dom=L

().

St(i) = { ISn| dom = L, (i) = i}. i

ISn

dom=L

() =iL

|St(i)| = |L| |St(i)| = k Ck1n1 (k 1)! =k

n Ckn k!.

i i L N Ckn ,

ISn

() =n

k=0

kn

Ckn2 k!.

iISn

() +1

n

ISn

def =n

k=0

k

n

Ckn2 k! +

nk=0

n k

n

Ckn2 k!

=n

k=0

k + n k

n

Ckn2 k! =

nk=0

Ckn2 k! = |ISn|.

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i i i 19

1) , k = n i H = G S(N). i, 5 i [2], |(ISn)| = 1 + n!m = 1 + r, m = |G|, r = [S(N) : G]. 2 ii i i r. 4 i [2] i ISn i , - i i i 1.

(ISn)

() +1

[ISn : H]

(ISn)

def =

(Sn)

() +1

r r = 1 +

(Sn)

().

4 i [2], i i i ISn i i ii H i i i i i S(N) i G.

(Sn)

()

. i S(N) i i ii S(N) i G .

(Sn)

() = 1 |(Sn)| = |Sn/ G| = r.

, ii (2) .2) 1 < k < n i G = S(M). i, 5 i [2]

|(ISn)| = 1 + (Ckn)

2 +n

i=k+1

(Cin)2 i!.

2 ii i i Ckn.

i i , M Cknk i N. 1 i ISn k ii i 1. i, i Ckni i.

, i i i k. X1 , i k M M. 3 i 4 t ISn k i (t) X1 i i i i, domt = M, i i (t) i X1 ii i i, domt = rant = M. t ISn k

i (t) i i Ckn i i def (t) =

Ckn 1. i ii i 1 i i i

k, i i i i i , X1 i i (t) i (ISn), domt = rant = M.

t ISn k + 1. ISn i Ckk+1 = k + 1 k, t i . (t) Ckk+1 = k + 1 i. ii i , t ISn i, k + 1 i n, (t) Cki i Ckn C

ki .

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20 ..

ii (2) :

1

Ckn

(ISn)

def =1

Ckn

Ckn +

Ckn2

Ckn 1

+

Ck+1n2

(k + 1)!

Ckn Ckk+1

+ . . . +

C

n1

n2

(n 1)!

C

k

n C

k

n1

+ (C

n

n)

2

(n)!

C

k

n C

k

n

= 1 + (Ckn)2 +

ni=k+1

(Cin)2i! Ckn

1

Ckn

ni=k+1

(Cin)2 i! Cki .

ii

(ISn)

(). i Ckn -

i . X1. St(X1) ={ (ISn)|X

1 = X1}. X1 i (t) i (ISn),

rank t = k. 3 i [2] i ISn i k -i ii i1. t i > k i X(t)1 = X1.

i 4 (M)t = M. I k! i i t ki -i M. i i k , i Ciknk , i t , Ciknk , i i-ii i (i k)! .

|St(X1)| = 1 +n

i=k+1

(Ciknk)2 (i k)! k!.

,

(ISn)

() = Ck

n |St(X1)| = Ck

n

1 +

n

i=k+1

(Cik

nk)2

(i k)! k!

.

i ,

(ISn)

() = Ckn +1

Ckn

ni=k+1

(Cin)2 i! Cki .

, ii (2) .3) 1 < k < n i G i S(M) i r.

5 i [2]

|(ISn)| = 1 + r (Ckn)

2 +n

i=k+1

(Cin)2 i!.

ii i i r Ckn. i , M Ckn k i N, i r . i Ckn i L1, . . . , LCkn r i i.

1 i ISn k iii 1. i i r Ckni i.

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i i i 21

, i i i k. X1 , i M M. 3 i 4 t ISn k i (t) X1 i i ii, domt = M, i i (t) i X1 i, domt = rant = M. t ISn k i-

(t) i r , iLi. i k, i i i i -i , r i i ii i 1, i ii Li i i i r i (ISn). G = A(M), r = 2 i i i i i, i . , k = 4 i G = K4, ii i r = 6 i , 6 i i (ISn) i 6 i. i i (ISn), domt = rant = M.

i , t ISn i, k + 1 i n (t) r Cki i r (Ckn C

ki ).

i i ii (2) :

1

rCkn

(ISn)

def =1

rCkn

rCkn + r2(Ckn)

2(Ckn) +n

i=k+1

(Cin)2i!r(Ckn C

ki )

= 1 + r(Ckn)2 +

ni=k+1

(Cin)2i! rCkn

1

Ckn

ni=k+1

(Cin)2 i! Cki .

ii

(ISn)(). i rCkn

i . X1. St(X1) = { (ISn)|X

1 = X1}. X1 i (t) i (ISn),

rank t = k. 3 i [2] i ISn i k - i ii i 1. t i > k iX

(t)1 = X1. i 4 (M

)t = M i tMx

Gx. i t iM |Gx| = |G| = m = k!

r. i i k , i -

Ciknk , i t , Ciknk , i

iii i (i k)! .

|St(X1)| = 1 +n

i=k+1

(Ciknk)2 (i k)!

k!

r.

,

(ISn)

() = rCkn |St(X1)| = rCkn

1 +n

i=k+1

(Ciknk)2 (i k)!

k!

r

= rCkn +1

Ckn

n

i=k+1

(Cin)2 i! Cki .

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22 ..

, ii (2) .4) 1 < k < n i G i S(M)

i r. 4 i [2] |(ISn)| = 1 +ni=k

(Cin)2 i!. 2, ii

i i r Ckn. i , M Ck

nk i N, i r

. i Ckn i L1, . . . , LCkn r i i.

i, 1 , i ISn k ii i 1. i, i rCkni i.

, i i i k. X1 , i M M. 3 i 4 t ISn k i (t) X1 i i ii, domt = M, i i (t) i X1 i,

domt = rant = M. t ISn k i(t) i r , i Li.

i i i , t ISn i, k +1 i n (t) r Cki i r (C

knC

ki ).

1

rCkn

(ISn)

def =1

rCkn

rCkn + (Ckn)

2k!r(Ckn 1) +n

i=k+1

(Cin)2i!r(Ckn C

ki )

= 1 + k!(Ckn)2 +

ni=k+1

(Cin)2i! k!Ckn 1Ckn

ni=k+1

(Cin)2 i! Cki .

i k, i i i i i , k! i i i ii i1. i ii Li i r i k! i i (ISn), i k M i . i ii , S(M) i i r i. i X1 iii t ISn k, X

(t)1 = X1, i

k!r

= |G| = m. ii

i t ISn i k, i , X(t)1 = X1, i . , ii i

(ISn)

() = rCkn

m +n

i=k+1

(Ciknk)2 (i k)! m

= k!Ckn +1

Ckn

ni=k+1

(Cin)2 i! Cki .

ii (2) i .

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i i i 23

i

1. .. i i i i i-

// i i. i i. 1998. . 2. . 16-21.

2. .. i i i i i-

ISn // i i. i: i.-. . 2006. . 1. . 9-16.

3. ., . . .: , 1972. . 1. 283 . . 2.

422 .

4. .. // . . . 1962. . 28, 3.

. 164-176.

i i ii i ,

, .

vtv [email protected]

i 20.05.2010

Voloshyna T.V. An analogue of Bernsides lemma for finite inverse symmetric semigroup , Car-

pathian Mathematical Publications, 2, 1 (2010), 1523.

An analogue of Bernsides lemma for transitive permutation representations of finite inverse

symmetric semigroup is obtained.

.. -

// . 2010. .2, 1. C. 1523.

.

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512+515.12

Gavrylkiv V.M.

ON REPRESENTATION OF SEMIGROUPS OF INCLUSION

HYPERSPACES

Gavrylkiv V.M. On representation of semigroups of inclusion hyperspaces, Carpathian Mathe-

matical Publications, 2, 1 (2010), 2434.

Given a group X we study the algebraic structure of the compact right-topological semi-

group G(X) consisting of inclusion hyperspaces on X. This semigroup contains the semigroup

(X) of maximal linked systems as a closed subsemigroup. We construct a faithful represen-tation of the semigroups G(X) and (X) in the semigroup P(X)P(X) of all self-maps of the

power-set P(X). Using this representation we prove that each minimal left ideal of (X) is

topologically isomorphic to a minimal left ideal of the semigroup pTpT, where by pT we denote

the family of pretwin subsets ofX.

Introduction

After discovering a topological proof of Hindman theorem [8] (see [10, p.102], [9]), topo-logical methods become a standard tool in the modern combinatorics of numbers, see [10],

[11]. The crucial point is that any semigroup operation defined on a discrete space X can

be extended to a right-topological semigroup operation on (X), the Stone-Cech compacti-

fication of X. The extension of the operation from X to (X) can be defined by the simple

formula

A B =

A X : {x X : x1A B} A

, (1)

where A, B are ultrafilters on X. Endowed with the so-extended operation, the Stone-

Cech compactification (X) becomes a compact right-topological semigroup. The algebraic

properties of this semigroup (for example, the existence of idempotents or minimal left ideals)have important consequences in combinatorics of numbers, see [10], [11].

The Stone-Cech compactification (X) of X is the subspace of the double power-set

P(P(X)), which is a complete lattice with respect to the operations of union and intersection.

In [7] it was observed that the semigroup operation extends not only to (X) but also to the

complete sublattice G(X) ofP(P(X)) generated by (X). This complete sublattice consists

of all inclusion hyperspaces over X.

2000 Mathematics Subject Classification: 20M30; 20M12; 22A15; 22A25; 54D35.Key words and phrases: binary operation, semigroup, right-topological semigroup, representation, self-linked

set, twin set, pretwin set, minimal left ideal.

c Gavrylkiv V.M., 2010

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On representation of semigroups of inclusion hyperspaces 25

By definition, a family F of non-empty subsets of a discrete space X is called an inclusion

hyperspace if F is monotone in the sense that a subset A X belongs to F provided

A contains some set B F. Besides the operations of union and intersection, the set

G(X) possesses an important transversality operation assigning to each inclusion hyperspace

F G(X) the inclusion hyperspace

F = {A X : F F (A F = )}.

This operation is involutive in the sense that (F) = F.

It is known that the family G(X) of inclusion hyperspaces on X is closed in the double

power-set P(P(X)) = {0, 1}P(X) endowed with the natural product topology. The induced

topology on G(X) can be described directly: it is generated by the sub-base consisting of

the sets

U+ = {F G(X) : U F } and U = {F G(X) : U F}

where U runs over subsets of X. Endowed with this topology, G(X) becomes a Hausdorff

supercompact space. The latter means that each cover of G(X) by the sub-basic sets hasa 2-element subcover. Let also N2(X) = {A G(X) : A A

} denote the family of all

linked inclusion hyperspaces on X and (X) = {F G(X) : F = F} the family of all

maximal linked systems on X.

By [6], both the subspaces (X) and N2(X) are closed in the space G(X). Observe that

U+ (X) = U (X) and hence the topology on (X) is generated by the sub-basis

consisting of the sets

U = {A (X) : U A}, U X.

The extension of a binary operation from X to G(X) can be defined in the same

manner as for ultrafilters, i.e., by the formula (1) applied to any two inclusion hyperspacesA, B G(X). In [7] it was shown that for an associative binary operation on X the space

G(X) endowed with the extended operation becomes a compact right-topological semigroup.

The structure of this semigroup was studied in details in [7]. In particular, it was shown

that for each group X the minimal left ideals of G(X) are singletons containing invariant

inclusion hyperspaces. Besides the Stone-Cech extension, the semigroup G(X) contains

many important spaces as closed subsemigroups. In particular, the space (X) of maximal

linked systems on X is a closed subsemigroup of G(X). The space (X) is well-known in

General and Categorial Topology as the superextension of X, see [12].

We call an inclusion hyperspace A G(X) invariant if xA = A for all x X. It follows

from the definition of the topology on G(X) that the set

G(X) of all invariant inclusion

hyperspaces is closed and non-empty in G(X). Moreover, the set

G(X) coincides with the

minimal ideal of G(X), which is a closed semigroup of right zeros. The latter means that

A B = B for all A, B

G(X).

The minimal ideal

G(X) contains the closed subset

N2(X) = N2(X)

G(X) of invariant

linked systems on X. The subset max

N2(X) of maximal invariant linked systems on X is

denoted by

(X). It can be shown that

(X) is a closed subsemigroup of

N2(X). By [2,

2.2], this semigroup has cardinality |

(X)| = 22|X|

for every infinite group X.

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26 Gavrylkiv V.M.

The thorough study of algebraic properties of semigroups of inclusion hyperspaces and

the superextensions of groups was started in [7] and continued in [1], [2] and [3]. In this

paper we construct a faithful representation of the semigroups G(X) and (X) in the semi-

group P(X)P(X) of all self-maps of the power-set P(X) and show that the image of (X) in

P(X)P(X) coincides with the semigroup (X,P(X)) of all functions f : P(X) P(X) that

are equivariant, monotone and symmetric in the sense that f(X \ A) = X \ f(A) for allA X. Using this representation we prove that each minimal left ideal of (X) is topo-

logically isomorphic to a minimal left ideal of the semigroup pTpT, where by pT we denote

the family of pretwin subsets of X. A subset A of a group X is called a pretwin subset if

xA X\ A yA for some x, y X.

1 Right-topological semigroups

In this section we recall some information from [10] related to right-topological semi-

groups. By definition, a right-topological semigroup is a topological space S endowed with

a semigroup operation : S S S such that for every a S the right shift ra : S S,

ra : x x a, is continuous. If the semigroup operation : S S S is (separately)

continuous, then (S, ) is a (semi-)topological semigroup.

From now on, S is a compact Hausdorff right-topological semigroup. We shall recall some

known information concerning ideals in S, see [10].

A non-empty subset I of S is called a left (resp. right) ideal if SI I (resp. IS I). If

I is both a left and right ideal in S, then I is called an ideal in S. Observe that for every

x S the set SxS = {sxt : s, t S} (resp. Sx = {sx : s S}, xS = {xs : s S}) is

an ideal (resp. left ideal, right ideal) ideal in S. Such an ideal is called principal. An ideal

I S is called minimalif any ideal ofS that lies in I coincides with I. By analogy we defineminimal left and right ideals ofS. It is easy to see that each minimal left (resp. right) ideal

I is principal. Moreover, I = Sx (resp. I = xS) for each x I. This simple observation

implies that each minimal left ideal in S, being principal, is closed in S. By [10, 2.6], each

left ideal in S contains a minimal left ideal.

We shall use the following known fact, see [3, Lemma 1.1].

Proposition 1.1. If a homomorphism h : S S between two semigroups is injective on

some minimal left ideal of S, then h is injective on each minimal left ideal of S.

2 The function representation of the semigroup G(X)

In this section given a group X we introduce the function representation : G(X)

P(X)P(X) of the semigroup G(X) in the semigroup P(X)P(X) of all self-maps of the power-

set P(X) of X. The semigroup P(X)P(X) endowed with the Tychonov product topol-

ogy is a compact right-topological semigroup naturally homeomorphic to the Cantor cube

({0, 1}X)P(X) = {0, 1}XP(X). The sub-base of the topology ofP(X)P(X) consists of the sets

x, A+ = {f P(X)P(X) : x f(A)},

x, A = {f P(X)P(X) : x / f(A)}.

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Given an inclusion hyperspace A G(X) consider the function

A : P(X) P(X), A(A) = {x G : x1A A}

called the function representation of A.

Proposition 2.1. A function : P(X) P(X) coincides with the function representationA of some (invariant) inclusion hyperspace A G(X) if and only if is

1) equivariant in the sense that (xA) = x(A) for any A X and x X;

2) monotone in the sense that (A) (B) for any subsets A B of X;

3) () = , (X) = X (and (P(X)) {, X}).

Proof. To prove the only if part, take any inclusion hyperspace A G(X) and consider

its function representation A.

It is equivariant because

A(xA) = {y X : y1xA A} = {xy : y1A A} = x A(A)

for any x X and A X.

Also it is monotone because

A(A) = {x G : x1A A} {x G : x1B A} = A(B)

for any subsets A B of X.

It is clear that A() = and A(X) = X.

If A is invariant, then for every A A we get A(A) = X and for each A P(X) \ A

we get A(A) = .

To prove the if part, fix any equivariant monotone map : P(X) P(X) with () =

and (X) = X and observe that the family

A = {x1A : A X, x (A)}

is an inclusion hyperspace with A = . If (P(X)) {, X}, then the inclusion hyper-

space A is invariant.

Remark 2.1. If X is a left-topological group and A is the filter of neighborhoods of the

identity element e of X, then the functional representations A and A have transparent

topological interpretations: for any subset A X the set A(A) coincides with the interior

of a set A X while A(A) with the closure of A in X!

The correspondence : A A determines a map : G(X) P(X)P(X) called the

function representation of the semigroup G(X).

Theorem 1. The function representation : G(X) P(X)P(X) is a continuous injective

semigroup homomorphism.

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28 Gavrylkiv V.M.

Proof. To check that is a semigroup homomorphism, take any two inclusion hyperspaces

X, Y G(X) and let Z = X Y. We need to check that Z(A) = X Y(A) for every

A X. Observe that

Z(A) = {z G : z1A Z} = {z G : {x G : x1z1A Y} X } =

= {z G : Y(z1A) X } = {z G : z1Y(A) X } = X(Y(A)).

To see that is injective, take any two distinct inclusion hyperspaces X, Y G(X).

Without loss of generality, X \ Y contains some set A X. It follows that e X(A) but

e / Y(A) and hence X = Y.

To prove that : G(X) P(X)P(X) is continuous we first define a convenient sub-base of

the topology on the spaces P(X) and P(X)P(X). The product topology ofP(X) is generated

by the sub-base consisting of the sets

x+ = {A X : x A} and x = {A X : x / A}

where x X. On the other hand, the product topology on P(X)P(X) is generated by thesub-base consisting of the sets

x, A+ = {f P(X)P(X) : x f(A)} and x, A = {f P(X)P(X) : x / f(A)}

where A P(X) and x X.

Now observe that the preimage

1(x, A+) = {A G(X) : x A(A)} = {A G(X) : x1A A} = (x1A)+

is open in G(X). The same is true for the preimage

1

(x, A

) = {A G(X) : x / A(A)} = {A G(X) : x1

A / A} = (X\ x1

A)

which also is open in G(X).

3 The semigroup (X, P(X)) and its projections (X, F)

Since for a group X the function representation : G(X) P(X)P(X) is an isomorphic

embedding, instead of the semigroup (X) we can study its isomorphic copy (X,P(X)) =

((X)) P(X)P(X). Our strategy is to study (X, P(X)) via its projections (X, F) onto

the faces P(X)F of the cube P(X)P(X), where F is a suitable subfamily ofP(X).

Given a subfamily F P(X) byprF : P(X)

P(X) P(X)F, prF : f f|F,

we denote the projection ofP(X)P(X) onto its F-face P(X)F. Let

F = prF : (X) P(X)F

and

(X, F) = F((X)) = prF((X,P(X)) = (prF )((X)).

Now we detect functions f : F P(X) belonging to the image (X, F). Let us call a

family F P(X)

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X-invariant if xF F for every F F and every x X;

symmetric if for each A F we get X\ A F.

Theorem 2. A function f : F P(X) defined on a symmetric X-invariant subfamily

F P(X) belongs to the image (X, F) = F((X)) if and only if

1) f is equivariant;

2) f is monotone;

3) f is symmetric in the sense that f(X\ A) = X\ f(A) for each A F.

Proof. To prove the only if part, take any maximal linked system L (X) and consider

its function representation f = L : P(X) P(X).

By Proposition 2.1, the function f is equivariant and monotone. Consequently, the

restriction f|F satisfies the items (1), (2). To prove the third item, take any set A F andobserve that

f(X\ A) = {x X : x1(X\ A) L} = {x X : X\ x1A L} =

= {x X : x1A / L} = X\ {x X : x1A L} = X\ f(A).

This completes the proof of the only if part.

To prove the if part, take any function f : F P(X) satisfying the conditions 1)3)

and consider the family

Lf = {x1A : A F, x f(A)}.

We claim that this family is linked. Assuming the converse, find two sets A, B F andtwo points x f(A) and y f(B) with x1A y1B = . Then yx1A X \ B and

hence yx1f(A) f(X \ B) = X \ f(B) by the properties 1)3) of the map f. Then

x1f(A) X \ y1f(B), which is not possible because the neutral element e of the group

X belongs to x1f(A) y1f(B).

Enlarge the linked family Lf to a maximal linked family L (X). We claim that

L|F = f. Indeed, take any set A F and observe that

f(A) {x X : x1A Lf} {x X : x1A L} = L(A).

To prove the reverse inclusion, observe that for any x X \ f(A) = f(X \ A) we get

x1(X\ A) = X\ x1A Lf L. Since L is linked, x1A / L and hence x / L(A).

A subfamily F P(X) is called -incomparable if for any subset A, B F the inclusion

A B implies the equality A = B. In this case each function f : F P(X) is monotone,

so the characterization Theorem 2 simplifies as follows.

Corollary 3.1. A function f : F P(X) defined on a -incomparable symmetric X-

invariant subfamilyF P(X) belongs to the image (X, F) = F((X)) if and only if f is

equivariant and symmetric.

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30 Gavrylkiv V.M.

A subfamily F P(X) is called -invariantifL(F) F for every maximal linked system

L (X). In this case (X, F) FF is a subsemigroup of the right-topological group FF of

all self-maps ofF.

Now we see that Theorem 1 implies

Proposition 3.1. For any -invariant subfamilyF P(X) the map

F = prF : (X) (X, F) FF

is a continuous semigroup homomorphism and (X, F) is a compact right-topological semi-

group.

4 Self-linked sets in groups

Our strategy in studying minimal left ideals of the semigroup (X) consists in findinga relatively small -invariant subfamily F P(X) such that the function representation

F : (X) (X, F) is injective on some (equivalently all) minimal left ideals of (X).

The first step in finding such a family F is to consider the family of self-linked sets in X.

Definition 4.1. A subset A of a group X is self-linked if xA yA = for all x, y X.

Self-linked sets in (finite) groups were studied in details in [1]. The following simple

characterization can be easily derived from the definitions.

Proposition 4.1. For a subset A X the following conditions are equivalent:

1) A is self-linked;

2) the family of shifts {xA : x X} is linked;

3) AA1 = X;

4) A belongs to an invariant linked system A

N2(X);

5) A belongs to a maximal invariant linked system A

(X) = max

N2(X).

The following proposition was first proved in [3, 4.1]. Here we present a short proof for

completeness.

Proposition 4.2. For any invariant linked system L0

N2(X) the upper set

L0 = {L (X) : L L0}

is a closed left ideal in (X).

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Proof. Let A, B (X) be maximal linked systems with L0 B. Then for every subset

L L0 we get

L =

xX

x(x1L) A L0 A B

which means that L0 A B.

To show that L0 is closed in (X), take any maximal linked system L (X) \ L0and find a set A L0 with A / L. Since L is maximal linked, X \ A L. Consequently,

(X\ A) is an open neighborhood of L that does not intersect L0.

Observe that any linked system L N2(X) extending an invariant linked system L0

N2(X) lies in the inclusion hyperspace L0 . It turns out that sets from L

0 \ L0 have a specific

structure described in the following theorem.

Theorem 3. For any maximal invariant linked system L0

(X) and any A L0 \ L0there are points a, b X such that aA X\ A bA.

Proof. Fix a subset A L0 \ L0. We claim that

aA A = (2)

for some a X. Assuming the converse, we would conclude that the family {xA : x X}

is linked and then the invariant linked system L0 {xA : x X} is strictly larger than L0,

which impossible because of the maximality of L0.

Next, we find b X with

A bA = X. (3)

Assuming that no such a point b exist, we conclude that for any x, y X the union xAyA =X. Then (X \ xA) (X \ yA) = X \ (xA yA) = , which means that the family

{X \ xA : x X} is linked and invariant. We claim that X \ A L0 . Assuming the

converse, we would conclude that X \ A misses some set L L0. Then L A and hence

A L0 which is not the case. Thus X\ A L0 and hence {X\ xA : x X} L

0 because

L0 is invariant. Since L0 {X \ xA : x X} is an invariant linked system containing L0,

the maximality of L0 guarantees that G \ A L0 which contradicts A L0 .

Unifying the equalities (2) and (3) we get the required inclusions

aA X\ A bA.

5 Twin and pretwin sets in groups

Having in mind the sets appearing in Theorem 3 we introduce the following two notions.

Definition 5.1. A subset A of a group X is called

a twin subset if X\ A = xA for some x X;

a pretwin subset if xA X\ A yA for some x, y X.

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32 Gavrylkiv V.M.

By T and pT we denote the families of twin and pretwin subsets of X, respectively.

Proposition 5.1. The familiespT andT are -invariant.

Proof. Take any maximal linked system L (X) and consider its function representation

f = L : P(X) P(X), which is equivariant, monotone, and symmetric according to

Theorem 2.To show that the family pT is -invariant, take any pretwin set A pT and find two

points x, y X with xA X \ A yA. Applying to those inequalities the monotone

equivariant symmetric function f we get

xf(A) = f(xA) f(X\ A) = X\ f(A) f(yA) = yf(A),

which means that f(A) is pretwin.

If a set A is twin, then X \ A = xA for some x X and then X \ f(A) = f(X \ A) =

f(xA) = xf(A), which means that f(A) is a twin set.

Propositions 5.1 and 3.1 imply that (X,T) and (X, pT) both are compact right-

topological semigroups. The importance of the family pT is explained by the following

Theorem 4. For every maximal invariant linked system L0

(X) the restriction pT|L0 :

L0 (X, pT) is a topological isomorphism of the compact right-topological semigroups.

Proof. Since pT is continuous and the semigroups (X) and (X, pT) are compact. It

suffices to check that the restriction pT|L0 is bijective.

To show that it is surjective, take any function f (X, pT), which is equivariant,

monotone, and symmetric according to Theorem 2.By the proof of Theorem 2, the family

Lf = {x1A : A pT, x f(A)}

is linked. We claim that so is the family L0 Lf. Assuming the opposite we could find

disjoint sets A Lf and B L0. Since A is pretwin, xA X \ A yA for some x, y X.

Now we see that

B X\ A yA X\ yB,

which is not possible as B is self-linked and hence meets its shift yB.

Now extend the linked family L0 Lf to a maximal linked family L (X) and showthat L|pT = f (repeating the argument of the proof of Theorem 2).

Next, we show that the restriction pT|L0 is injective. Take any two distinct maximal

linked systems X, Y L0. It follows that there is a set A X \ Y . This set belongs to

L0 \L0 and hence is pretwin by Theorem 3. Now the definition of the function representation

yields that e X(A) \ Y(A), witnessing that pT(X) = pT(Y).

Since the function representation pT is injective on the left ideal L0 of (X), it is

injective on some minimal left ideal of (X) and hence is injective on each minimal left ideal

of (X), see Proposition 1.1. In such a way we prove

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Corollary 5.1. The function representation pT : (X) (X, pT) is injective on each

minimal left ideal of (X). Consequently, each minimal left ideal of (X) is topologically

isomorphic to a minimal left ideal of the semigroup (X, pT).

6 Acknowledgments

The author express his sincere thanks to Taras Banakh for help during preparation of

the paper.

References

1. Banakh T., Gavrylkiv V., Nykyforchyn O. Algebra in superextensions of groups, I: zeros and commuta-

tivity, Algebra Discrete Math, 3 (2008), 1-29.

2. Banakh T., Gavrylkiv V. Algebra in superextension of groups, II: cancelativity and centers, Algebra

Discrete Math, 4 (2008), 1-14.

3. Banakh T., Gavrylkiv V. Algebra in the superextensions of groups, III: minimal left ideals, Mat. Stud.,

31, 2 (2009), 142-148.

4. Bilyeu R.G., Lau A. Representations into the hyperspace of a compact group, Semigroup Forum 13 (1977),

267-270.

5. Engelking R. General Topology, PWN, Warsaw, 1977.

6. Gavrylkiv V. The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28, 1 (2007),

92-110.

7. Gavrylkiv V. Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud., 29, 1 (2008),

18-34.

8. Hindman N., Finite sums from sequences within cells of partition of N, J. Combin. Theory Ser. A 17

(1974), 1-11.

9. Hindman N., Ultrafilters and combinatorial number theory, Lecture Notes in Math. 751 (1979), 49-184.

10. Hindman N., Strauss D. Algebra in the Stone-Cech compactification, de Gruyter, Berlin, New York,

1998.

11. Protasov I. Combinatorics of Numbers, VNTL, Lviv, 1997.

12. Teleiko A., Zarichnyi M. Categorical Topology of Compact Hausdorff Spaces, VNTL, Lviv, 1999.

13. Trnkova V. On a representation of commutative semigroups, Semigroup Forum, 10, 3 (1975), 203-214.

Vasyl Stefanyk Precarpathian National University,

Ivano-Frankivsk, Ukraine.

[email protected]

Received 25.05.2010

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34 Gavrylkiv V.M.

i .. i ii // i -

i ii. 2010. .2, 1. C. 2434.

i i i

G(X), i i ii i X. i

i i (X) i ii.

i G(X) (X) ii P(X)P(X) i i-

i- P(X) . , ii i i i (X) i i ii i

i i pTpT.

.. // -

. 2010. .2, 1. C. 2434.

-

G(X), X.

(X)

. G(X) (X)

P(X)P(X) - P(X) .

, (X) pTpT.

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517.98

Hryniv R.O.

ANALYTICITY AND UNIFORM STABILITY IN THE INVERSE

SPECTRAL PROBLEM FOR IMPEDANCE STURMLIOUVILLE

OPERATORS

Hryniv R.O. Analyticity and uniform stability in the inverse spectral problem for impedance

SturmLiouville operators, Carpathian Mathematical Publications, 2, 1 (2010), 3558.

We prove that the inverse spectral mapping reconstructing the impedance function of theSturmLiouville operators on [0, 1] in impedance form from their spectral data (two spectra or

one spectrum and the corresponding norming constants) is analytic and uniformly stable in a

certain sense.

1 Introduction

The main goal of this paper is to establish analyticity and uniform continuity of solutions

to the inverse spectral problems for a certain class of SturmLiouville operators on [0, 1] inthe so-called impedance form. Namely, the spectral problems of interest are

(a2(x)y(x)) = a2(x)y(x), x [0, 1], (1)

subject to suitable boundary conditions, e.g., the Neumann ones

y(0) = y(1) = 0 (2)

or NeumannDirichlet ones

y(0) = y(1) = 0. (3)

Here a > 0 is an impedance function, which will be supposed to belong to the Sobolev space

W12 (0, 1), so that the logarithmic derivative := (log a) (called the logarithmic impedance

below) is in L2(0, 1). Without loss of generality we may assume that a(0) = 1, so that

a(x) = expx

0(s) ds

. Such spectral problems arise in many applications, e.g., in modelling

propagation of sound waves in a duct [44], torsional vibrations of the earth [17] or longitudinal

vibrations in a thin straight rod [13].

2000 Mathematics Subject Classification: Primary 34A55, Secondary 34L40, 47E05.Key words and phrases: inverse spectral problems, SturmLiouville operators, impedance, analyticity, uni-

form stability.

c Hryniv R.O., 2010

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36 Hryniv R.O.

The corresponding differential operators SN and SD given by the differential expression

(y) := a2(a2y) and boundary conditions (2) and (3) respectively are self-adjoint in the

weighted Hilbert space L2

(0, 1); a2 dx

and have simple discrete spectra accumulating at

+. We denote by 0 = 0 < 1 < the eigenvalues of SN and by 0 < 0 < 1 < those of SD. The inverse spectral problem is to reconstruct the impedance function a or its

logarithm from the spectra of SN and/or SD.For the standard SturmLiouville operators, i.e., those generated by the differential ex-

pression

d2

dx2+ q,

with q a real-valued locally integrable potential, it was proved by Borg [7] in 1946 that,

generically, knowledge of the spectrum corresponding to one set of boundary conditions (e.g.

Neumann ones or NeumannDirichlet ones) does not allow to unambiguously determine q.

(An exceptional situation where this is possible was pointed out by Ambartzumyan [5] in

1929.) However, two such spectra do uniquely determine q.

The same holds true for the inverse spectral problem of reconstructing the impedance

function a of the operators SN or SD. In fact, these operators are unitarily equivalent to self-

adjoint operators TN and TD acting in L2(0, 1) and generated by the differential expression

() := 1a

d

dxa2

d

dx

1

a=

ddx

+ d

dx

(4)

and the boundary conditions

y[1](0) = y[1](1) = 0 (5)

and y[1](0) = y(1) = 0 (6)

respectively. Here and hereafter f[1](x) := f(x) (x)f(x) shall denote the quasi-derivativeof a function f. Moreover, for a W22 (0, 1) the differential expression () can be recast inthe potential form

() = d2

dx2+ + 2

with potential q = + 2. For a W12 (0, 1) the reduction to the potential form is stillpossible, but the potential q becomes a distribution from W12 (0, 1) [39]. SturmLiouville and

Schrodinger operators with singular potentials (that are, e.g., point interactions, measures,or distributions) have been widely studied; we refer the reader, e.g., to the books [1, 3]

and to review paper [40] where additional references can be found. Inverse problems for

distributional potentials in the space W12 (0, 1) have also been successfully treated; see,

e.g., [24,41].

This suggests the following method of solving the inverse spectral problem for impe-

dance SturmLiouville operators under consideration: first, one recasts the problem (1) in

the potential form, then uses one of the algorithms reconstructing the potential q from

the spectral data

(n), (n)

of TN and TD, and, finally, finds by solving the Riccati

differential equation + 2 = q. However, this equation may not possess global solutions on

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Uniform stability in the inverse problem 37

[0, 1], whence it is desirable to find a way to reconstruct the impedance a or its logarithmic

derivative directly from the spectral data for the operators TN and TD.

In the papers [2, 6, 8, 32, 35] several approaches to reconstruction of the impedance a W12 (0, 1) were suggested and the corresponding spectral data were completely described.

These necessary and sufficient conditions require that the spectra (n) and (n) must

(i) interlace, i.e., that n < n < n+1 for all n Z+, and

(ii) satisfy the asymptotic relationsn = n + 2n,

n = (n +

12

) + 2n+1,

where the sequence (n) belongs to 2.

Moreover, the induced mapping from the spectral data

(n), (n)

into the impedance

function a providing a solution to the inverse spectral problem was shown in [6] and [32] to

be locally continuous in a certain sense. In particular, this yields local stability of the inversespectral problem; see also similar stability results for the related problem of reconstructing

the potential q in [4, 7, 16, 1921, 31, 33, 34, 3638, 46]. Here we introduce a metric on the

set of the spectral data

(n), (n)

by e.g. identifying such data with the sequence (n) in

the representation of item (ii) above. Typically, this local stability states that, for a fixed

M > 0, there are positive and L with the following property: if potentials q1 and q2(resp., logarithmic impedances 1 and 2) are such that q1 M and q2 M (resp.,1 M and 2 M) and the corresponding spectral data 1 :=

(1,n), (1,n)

and

2 :=

(2,n), (2,n)

satisfy 1 2 , then

q1 q2 L1 2 (7)(resp., then

1 2 L1 2) (8)for a suitable norm . For instance, local stability results with respect to the L2(0, 1)-normwere established in [32, 38] in the regular case q L2(0, 1), and in [6,8,32] for impedanceSturmLiouville operators. In [16,33] the case L(0, 1) was treated; earlier Hochstadt in [20,

21] proved stability if only finitely many eigenvalues in one spectrum are changed. The

papers [19,36] studied to what extent only finitely many eigenvalues in one or both spectra

determine the potential, and the latter problem in the non-self-adjoint setting was recentlydiscussed in [31]. Also, stability of the inverse spectral problems on semi-axis was proved

in [30,37], and the inverse scattering problem on the line was studied in [10,18].

However, the above results cannot be considered satisfactory, as they refer to the norm

of the potential q (resp. of the logarithmic impedance ) to be recovered and thus specify

neither the allowed noise level nor the Lipschitz constant L. Therefore we need a global

stability result that asserts (7) whenever the spectral data 1 and 2 run through bounded

sets N and with L only depending on N.

Recently, such a uniform stability in the inverse spectral problem for SturmLiouville

operators on [0, 1] was established by Shkalikov and Savchuk [43]. They considered operators

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38 Hryniv R.O.

with real-valued potentials from the Sobolev spaces Ws2 (0, 1) with s > 1. (For negatives, such potentials are distributions; see [40] for the review on SturmLiouville operators

with distributional potentials.) Their approach for solving the inverse spectral problem was

based on the so called Prufer angle and used extensively the implicit function theorem. In our

work [22] analyticity and global stability of the inverse spectral mapping for s [1, 0] wasestablished using a different approach that generalizes the classical method due to Gelfandand Levitan [12] and Marchenko [29] and has been successfully applied to reconstruction of

SturmLiouville operators with singular potentials in [24,25].

The main aim of this paper is to prove analyticity and Lipschitz continuity on bounded

subsets of the inverse spectral mapping

(n), (n) for the class of the SturmLiouville

operators in impedance form with logarithmic impedance L2(0, 1). To this end we usethe approach of [2] to the inverse spectral problem for impedance SturmLiouville operators

based on the Krein equation [27] and further develop the methods of [22]. Also, we discuss the

analogous properties in the inverse spectral problem of reconstruction of from the Neumann

spectrum (n) and the corresponding norming constants n defined in Subsection 2.1.

We mention that the methods of [2] could be used to treat logarithmic impedances

belonging to Lp(0, 1) with p [1, ). However, apart from some technicalities caused bymore complicated properties of the Fourier transform in Lp(0, 1) for p = 2, the approachwould remain the same and we decided to sacrifice the generality to simplicity of presentation.

See Section 5 for discussion of possible generalizations.

The paper is organised as follows. In the next section, we state the main results of the

paper and recall the method of reconstructing the impedance SturmLiouville operators from

their spectral data using the GelfandLevitanMarchenko and Krein equations. In Section 3,

we show analyticity and uniform continuity in the inverse problem of reconstructing the

logarithmic impedance from the spectrum of the operator TN() and the sequence of thecorresponding norming constants. Reconstruction from two spectra (those of TN() and

TD()) is discussed in Section 4; there the problem is reduced to the one studied in Section 3

by showing that the norming constants depend analytically and Lipschitz continuously on

these spectra. The last Section 5 discusses some ways of extending the results to a wider class

of operators. Finally, three appendices contain auxiliary results on some related nonlinear

mappings in L2(0, 1), on relation between some analytic functions of sine type and their

zeros, and on the special Banach algebra that were used in the proofs.

2 Preliminaries and main results

In this section we state the main results of the paper and recall the method of solution of

the inverse spectral problem based on the GelfandLevitanMarchenko [28] and Krein [27]

equations. All the missing details can be found in [2].

2.1 Spectral data

Throughout this subsection, designates a fixed real-valued function in L2(0, 1). We denote

by n and n, nZ+, the eigenvalues of the operators TN() and TD() respectively defined

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via (4)(6) and recall that these eigenvalues interlace, i.e., n < n < n+1 for all n Z+,and satisfy the relations

n = n + 2n,

n = (n +

12

) + 2n+1 (9)

with some 2(Z+)-sequence = (n).

For = 2 C, the equation ()u = 2u subject to the initial conditions u(0) = 1 andu[1](0) = 0 has the solution

c(x, ) = cos x +

x0

k(x, t)cos tdt, (10)

where k is the kernel of the so called transformation operator. Clearly, cos x is a solution

of the unperturbed equation (0)u = 2u with = 0; it is mapped into the solution c(, )for a generic by means of the transformation operator via (10). The function k vanishes

for a.e. (x, t) [0, 1]2

with x < t and, for every x [0, 1], k(x, ) belongs to L2(0, 1) andthe mapping x k(x, ) is continuous from [0, 1] into L2(0, 1). Also, there exists a kernel k1with similar properties such that

c[1](x, ) = sin x x0

k1(x, t)sin tdt; (11)

we recall that f[1] := f f is the quasi-derivative of a function f.Set 2n :=

n and 2n+1 :=

n, n Z+. Then c(, 2n) is an eigenfunction of

the operator TN() corresponding to the eigenvalue n = 22n, and we call the number

1

n := 1/(2c(, 2n)2

) the norming constant for this eigenvalue. It is known [2] that

n = 1 + n, (12)

where the sequence := (n)nZ+ belongs to 2. Moreover, the norming constants ncan be determined from the spectra of the operators TN() and TD() as follows. We set

C() := c(1, ) and S() := c[1](1, ); due to (10) and (11) these are entire functions of

exponential type 1 with zeros n and

n respectively. The Hadamard canonical

products for S and C are

S() = 2

n=1

22n 22n2

, C() =

n=0

2n+1 22(n + 1

2)2

, (13)

so that S and C are uniquely determined by their zeros. Then we have (cf. [2])

n = 2nS(2n)C(2n)

, (14)

where the dot denotes the derivative in .

Here and hereafter,f

shall stand for the L2(0, 1)-norm of a function f.

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40 Hryniv R.O.

2.2 The main results

We introduce the set N of pairs

(n)nZ+, (n)nZ+

with the following properties:

the sequences (n) and (n) strictly interlace, i.e., n < n < n+1 for all n Z+;

the sequence := (k)kZ+, with 2n :=

n n and 2n+1 := n (n +1

2),belongs to 2.

In this way every element :=

(n), (n)

ofN is identified with a sequence (n) in 2thus inducing a metric on N. Namely, if1 and 2 are elements ofN and 1 := (1,n) and

2 := (2,n) are the corresponding 2-sequences of remainders, then

distN(1,2) := 1 22.

In what follows, 0 shall stand for the element ofN corresponding to = 0; then we get

distN(,0) =

(n)

2 .

According to [2], every element ofN gives the eigenvalue sequences of the operators

TN() and TD() for a unique real-valued function L2(0, 1) and, conversely, for everyreal-valued L2(0, 1) the spectra of the corresponding SturmLiouville operators TN()and TD() form an element ofN. When the logarithmic impedance varies over a bounded

subset of L2(0, 1), then the corresponding spectral data

(n), (n)

remain in a bounded

subset ofN. Moreover, the Prufer angle technique (cf. [41,42]) yields then a positive d such

that all the corresponding spectral data

(n), (n)

are d-separated, i.e., that n n dand n+1 n d for every n Z+. Summarizing, we conclude that the uniform stabilityof the inverse spectral problem we would like to establish is only possible on bounded sets

of spectral data in N that are d-separated for some d > 0.This motivates the following definition.

Definition 2.1. For d (0, /2) and r > 0, we denote byN(d, r) the set of all Nthat are d-separated and satisfy distN(,0) r.

In these notations, the first main result of the paper reads as follows.

Theorem 1. For every d (0, /2) and r > 0, the inverse spectral mapping

N(d, r)

L2(0, 1) (15)

is analytic and Lipschitz continuous.

See [9] for analyticity of mapping between Banach spaces. In fact, as in [22], we prove

first the analyticity and Lipschitz continuity of the inverse spectral problem of reconstructing

from the Neumann spectrum (n) and the norming constants (n) (see Theorem 2 below),

and then derive Theorem 1 by showing that the norming constants depend analytically and

Lipschitz continuously on the two spectra.

More exactly, we denote by L the family of strictly increasing sequences := (n)nZ+such that 2n :=

n

n form an element of2 and pull back the topology on L from that

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of 2 by identifying such with (2n) 2. For d (0, ) and r > 0, we denote by L(d, r)the closed convex subset ofL consisting of sequences (n)nZ+ such that (2n)2 r andn+1 n d for all n Z+. Next, we write A for the set of sequences := (n)nZ ofpositive numbers such that the sequence (n) with n := n 1 belongs to 2. This inducesthe topology of 2 on A; we further consider closed subsets A(d, r) ofA consisting of all

(n) satisfying the inequalities n d for all n Z and the relation (n)2 r.It is known [2] that, given an element (,) L A, there is a unique real-valued

L2(0, 1) such that is the sequence of eigenvalues and the sequence of normingconstants for the SturmLiouville operator TN(). Some further properties of the induced

mapping are described in the following theorem.

Theorem 2. For every d (0, ) and d (0, 1) and every positive r and r, the inversespectral mapping

L(d, r) A(d, r) (,) L2(0, 1)


2.3 Solution of the inverse spectral problem using the Krein equa-

tion

The classical algorithm of reconstructing the potential q = + 2 of a SturmLiouville op-

erator uses the so called GelfandLevitanMarchenko (GLM) equation relating the spectral

data (,) and the transformation operator K, see e.g. the monographs [28,29] for details.

The derivation of the GLM equation sketched below follows the reasoning of [24], to which

we refer the reader for further details.First we notice that due to the asymptotics of n and n the series in

h(s) := 1 + 2

n=0

n cos(22ns) cos(2ns)

(16)

converges in L2(0, 1) (in fact, h is an even function on (1, 1)). Next, denote by F an integraloperator in L2(0, 1) with kernel

f(x, t) := 12hx+t2 + hxt2 . (17)

Starting with the resolution of identity for the operator TN(),

I = 2

n=0

n( , cn)cn,

with cn = c( , 2n) being the eigenfunction corresponding to the eigenvalue n = 22n, andusing the relations (10) and the definition of F, after straightforward transformations one

arrives at the equality

I = (I + K)(I+ F)(I+ K).

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42 Hryniv R.O.

Actually, the above equality rewritten in terms of the kernels k and f of the operators K

and F produces the GLM equation,

k(x, t) + f(x, t) +

x0

k(x, s)f(s, t) ds = 0, x > t. (18)

Given the spectral data and thus the kernel f, one solves the GLM equation for the kernelk and then determines the potential q from the relation

q(x) = 2d

dxk(x, x). (19)

However, this approach does not work for impedance SturmLiouville operators under

consideration since formula (19) is then meaningless: indeed, the kernel k is not regular

enough to have a well-defined restriction k(x, x) to the diagonal and the potential q = + 2

is a distribution rather then a regular function. Instead, one can use the method of Krein

that reconstructs the function

L2(0, 1) directly. The original method was suggested

by Krein [27] for smooth functions and was further developed for the class of impedanceSturmLiouville operators with Lp(0, 1), p [1, ) in [2].

Namely, with the function h of (16), one considers a different GLM-type integral equation

(called the Krein equation)

r(x, t) + h(x t) +x0

r(x, s)h(s t) ds = 0, 0 < t < x < 1, (20)

of which the GLM equation (18) is the even part (in the sense that if r is a solution to (20),

then the function

k(x, t) := 12

r(x, xt2 ) + r(x, x+t2 )

solves (18)). It can be proved (see the next section) that equation (20) possesses a unique

solution r and, moreover, the function satisfies the equality

= r(, 0). (21)

This formula will be the basis of the reconstruction algorithm and stability analysis.

3 Stability of the inverse spectral problem: norming constants

In this section, we prove Theorem 2 on analytic and Lipschitz continuous dependence of

the logarithmic potential determining the impedance SturmLiouville operator TN() on

its eigenvalues n and norming constants n.

We shall study the correspondence between the data (,) L(d, r) A(d, r) andthe functions through the chain of mappings

(,) h r ,

in which h is the function of (16), r is the kernel solving the Krein equation (20), and, finally,

is given by (21).

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Lemma 3.1. The mapping

L(d, r) A(d, r) (,) h L2(0, 1)


Proof. We have h = 1 + h + h,, where

h(s) := 2

n=0

[cos(22ns) cos(2ns)], h,(s) := 2

n=0

n cos(22ns);

recall that the numbers 2n = 2n n and n := n 1 form sequences in 2 that inducethe topology ofL and A.

Introduce the function f L2(0, 1) whose Fourier coefficients are f(0) = 0 and

f(n) =

f(n) := 2n

for n N; then we have h = 1(f) with the mapping 1 of Lemma A.1. Therefore thefunction h depends analytically and Lipschitz continuously on f in bounded sets. Since

the mapping sending (2n) 2 into f L2(0, 1) is linear and quasi-isometric in the sensethat f =

2(2n), we conclude that the mapping h is analytic and Lipschitz

continuous on bounded sets.

Next, let g be the function in L2(0, 1) whose Fourier coefficients are

g(n) = g(n) := n

for n Z+. Then h, = 2(f, g) with 2 being the mapping of Lemma A.2. Theproperties of 2 and of the mapping (n) g then establish the required dependenceof h, on (,). The lemma is proved.

Solubility of the Krein equation crucially relies on the following property of the convolu-

tion operator H = H(,) defined via

(Hf)(x) :=

1

0

h(x t)f(t) dt,

with the function h of (16).

Lemma 3.2. For every d (0, ), d (0, 1), and positive r and r, there exists > 0 withthe following property: if (,) is an arbitrary element ofL(d, r) A(d, r) and h is thefunction of (16), then for the corresponding convolution operator H we have I+ H I.

Proof. Observing that

1

0

cos2n(x

t)f(t) dt = cos 2nx

1

0

cos2ntf(t) dt + sin 2nx 1

0

sin2ntf(t) dt

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44 Hryniv R.O.

and that the functions 1,

2sin2nx,

2cos2nx, n N, form an orthonormal basis ofL2(0, 1), we find that

((I + H)f, f) = (f, f) + 2 limk

k

n=0n

|(f, cos22ns)|2 + |(f, sin22ns)|2]

|(f, 1)|2 2 limk

kn=1

|(f, cos2ns)|2 + |(f, sin2ns)|2]= 2

n=0

n|(f, cos22ns)|2 + |(f, sin22ns)|2]

= 20|(f, 1)|2 +

n=1

n|(f, e22nis)|2 + |(f, e22nis)|2].

It follows from the results of [14, Ch. VI], [45, Ch. 4] that the system

E :=

e22nis}nN {1}

e22nis}nN

is a Riesz basis of L2(0, 1). Moreover, it was shown in [23] that there exists m = m(d, r) > 0

that gives a lower bound of E for every L(d, r). Since the inclusion A(d, r)implies that n d for all n Z, we get

((I + H)f, f) = 20|(f, 1)|2 +

n=1

n|(f, e22nis)|2 + |(f, e22nis)|2] dmf2,

and the proof is complete.

To study solubility of the Krein equation (20), we shall regard it as a relation between

the corresponding integral operators. To this end we recall several notions that will be used.

The ideal S2 of HilbertSchmidt operators in L2(0, 1) consists of integral operators whose

kernels are square integrable on := [0, 1] [0, 1]. The linear set S2 becomes a Hilbertspace under the scalar product

A, B2 := tr(AB) :=10

10

a(x, y)b(x, y) dxdy,

where a and b are the kernels of A and B respectively; in particular, AS2 := A, A1/22 isthe corresponding norm.

As an example, the inequality

10

10

|h(x y)|2 dxdy 2h2

implies that the convolution operator H belongs to S2 and, moreover, H2S2 2h2.Denote by S+2 the subspace ofS2 consisting of all HilbertSchmidt operators with lower-

triangular kernels. In other words, AS2 belongs to S

+

2

if the kernel a of A satisfies

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a(x, y) = 0 for a.e. 0 x < y 1. For an arbitrary A S2 with kernel a the cut-off a+ ofa given by

a+(x, y) =

a(x, y) for x y,0 for x < y

generates an operator A+

S+2 , and the corresponding mapping

P+ : A

A+ turns out to

be an orthoprojector in S2 onto S+2 , i.e. (P+)2 = P+ and P+A, B2 = A, P+B2 for allA, B S2; see details in [15, Ch. I.10].

With these notations, the Krein equation (20) can be recast as

R + P+H+ P+(RH) = 0 (22)

or

(I+ P+H)R = P+H,where P+X is the linear operator in S2 defined by P+X Y = P+(Y X) and I is the identityoperator in S2. Therefore solubility of the Krein equation and continuity of its solutionson H is strongly connected with the properties of the operator P+H.

Lemma 3.3. For every X B, the operator P+X is bounded in S2. Moreover, for everyconvolution operator H from the set

H := {H = H(,) | (,) L(d, r) A(d, r)} S2

the operatorI+ P+H is invertible in B(S+2 ) and the inverse (I+ P+H)1 depends analyticallyand Lipschitz continuously on H H in the topology ofS2.

Proof. Boundedness ofP+X is a straightforward consequence of the inequality

P+X YS2 Y XS2 XBYS2 ,

cf. [14, Ch. 3]. Assume next that I+ X I in L2(0, 1); then for Y S+2 we find that

(I+ P+X )Y, Y2 = Y, Y2 + Y X , Y 2 = tr

Y(I + X)Y

.

Since Y(I+ X)Y Y Y and the trace is a monotone functional, we get

(I+ P+X )Y, Y2 Y, Y2,

i.e., I+ P+X Iin S+2 .Applying now Lemma 3.2, we conclude that for every H H it holds I+ P+H Iwith

of that lemma depending only on d, d, r, and r; therefore, I+ P+H is boundedly invertiblein B(S+2 ) and (I+ P+H)1B(S+

2) 1.

Since P+H depends linearly on H, it follows that the mapping H (I+ P+H)1 from S2 intoB(S+2 ) is analytic and Lipschitz continuous on the set H. The proof is complete.

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46 Hryniv R.O.

Corollary 3.1. For every H H, the Krein equation (22) has a unique solutionR := (I+ P+H)1P+H S+2 ;

moreover, R depends analytically and Lipschitz continuously in S+2 on H H S2.It follows that the kernel r(x, t) of R is square integrable in the domain and depends

analytically and Lipschitz continuously in L2() on H. However, we need to know that

r( , 0) is well defined and belongs to L2(0, 1).To this end we use the Krein equation to find that

r(x, t) = h(x t) 10

r(x, s)h(s t) ds

as a function of x depends continuously in L2(0, 1) on t [0, 1]. Indeed, since the shiftf() f( t) is a continuous operation in L2(R), h( t) enjoys the required property.Next, since the kernels r and h belong to L2(), we find that1

0

10

r(x, s)h(s t) ds2 dx

10

dx

10

|r(x, s)|2 ds10

|h(s t)|2 ds

210

|h(s)|2 ds10

10

|r(x, s)|2 dsdx < .

(23)

Thus the function

1

0

r(x, s)h(s t) ds (24)

of the variable x [0, 1] belongs to L2(0, 1); moreover, continuity of the shifts h( t) andestimate (23) show that function (24) depends continuously in L2(0, 1) on t [0, 1]. Wethus conclude that indeed r(, t) depends continuously in L2(0, 1) on t [0, 1]. In particular,r(x, 0) is a well-defined function in L2(0, 1).

Finally, we again use the Krein equation and (21) to get the relation

(x) = r(x, 0) = h(x) 10

r(x, s)h(s) ds.

The integral on the right-hand side is a bilinear expression in h and r. In view of the analytic

dependence of r on h stated in Corollary 3.1 and estimates (23), this yields analyticity andLipschitz continuity of r(x, 0) on h L2(0, 1). On account of Lemma 3.2, the proof ofTheorem 2 is complete.

4 Reconstruction from two spectra

We recall that the norming constants n for the SturmLiouville operator TN() can be

determined from the spectra (n) and (n) of TN() and TD() by the formula (14),

n =

2n

S(2n)C(2n)

,

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where the entire functions S and C are given by the canonical products (13) over n = 22n

and n = 22n+1 respectively. This induces a mapping from the spectral data :=

(n), (n) N into the norming constants := (n) A. In this section, we shall

establish Theorem 1 by proving the following result.

Theorem 3. For every d

(0, /2) and r > 0, the mapping

N(d, r) A (25)

is analytic and Lipschitz continuous; moreover, there exist positive constants d and r such

that the range of this mapping belongs toA(d, r).

By definition, A consists of elements of the commutative unital Banach algebra A intro-

duced in Appendix C. We observe that the metrics on A agrees with the norm of A, and

thus the results of Appendix C yield the following statement.

Proposition 4.1. For every positive d and r, the set A(d, r) consists of invertible elementsofA. Moreover, the mapping 1 is analytic and Lipschitz continuous in A on A(d, r),and its range lies in A

(1 + r)1, rd1

.

In view of Proposition 4.1, it suffices to prove Theorem 3 with replaced by 1.

The elements of the sequence 1 are 1n = S(2n)C(2n)/2n. We shall show that thesequences

:=

(1)n+1S(2n)/2n

nZ+, :=

(1)nC(2n)

nZ+

form elements ofA. Thus Theorem 3 will be proved if we show that the mappings

N(d, r) A, N(d, r) A (26)

enjoy the properties required therein for the mapping (25).

To begin with, integral representations (10) and (11) of the solution c( , ) and its quasi-derivative c[1]( , ) yield the formulae

S() = sin 10

k1(1, t)sin tdt, (27)

C() = cos + 1

0

k(1, t)cos tdt (28)

for the functions S and C. Therefore both expressions S(2n)/2n and C(2n) can berecast in the form

cos 2n +

10

g(t)cos 2nt dt

with g(t) = tk1(1, t) for the former expression and g(t) = k(1, t) for the latter. The sequences

and have therefore similar structures; namely, their n-th element equals

cos 2n + (

1)n

1

0

g(t)cos 2nt dt (29)

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48 Hryniv R.O.

for respective g; here, as usual, 2n := 2n n.Clearly, the mapping (2n) (cos 2n 1) is analytic in 2. Its Lipschitz continuity

follows from the inequality | cos x cos y| |x y|; also, the inequality 1 cos x x2/2yields the estimate

(cos 2n 1)2 12(2n)22. (30)Set

g(s) :=

g(1 2s), s [0, 1

2),

g(2s 1), s [12

, 1];

then straightforward transformations give

vn := (1)n10

g(t)cos 2nt dt = (1)n10

g(s)ei2n(12s) ds

=

1

0

g(s)ei2n(12s)e2ins ds.

(31)

Therefore the above number vn gives the n-th Fourier coefficient of the function u := (f, g),

where is the mapping of Lemma A.3 and f is the function introduced in the proof of

Lemma 3.1. It follows from Lemma A.3 that the sequence (u(n))nZ of Fourier coefficients

ofu depends analytically and boundedly Lipschitz continuously in 2 on f and g. We prove

in the lemma below that the functions k(1, ) and k1(1, ) (and thus the correspondingtransformates g) depend in the same manner on = (,) N(d, r).

Lemma 4.1. The mappings

N(d, r) (,) k(1, ) L2(0, 1),

N(d, r) (,) k1(1, ) L2(0, 1)

are analytic and Lipschitz continuous.

Proof. Since both mappings can be treated similarly, we only consider the second one. By

definition, we have S(2n)/2n = 0, and thus the numbers 2n = n + 2n, n Z, arezeros of the odd entire function S()/ of (27). The required properties of the mapping

k1(1, ) follow now from the results of [26]; see Appendix B.

The above reasoning justifies the inclusion1

A

as well as analyticity and Lipschitzcontinuity of the mappings of (26). It remains to prove that there exist positive d and r

such that, for every N(h, r), the corresponding elements and belong to A(d, r).Existence of such an r follows from the uniform estimates of the 2-norms of the sequences

(cos 2n 1) of (30) and the fact thatnZ+

|vn|2 (f, g)2,

see (31) and the discussion following it. Indeed, in view of Lemma A.3 the function u =

(f, g) remains in the bounded subset ofL2(0, 1) when f and g vary over bounded subsets

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of L2(0, 1), and the latter is the case when runs over N(d, r) by the definition of the

functions f and g and Lemma 4.1.

Next, in view of formula (13) and the interlacing property of n and n, the numbers

n = (1)nS(2n)/2n and n = (1)nC(2n) are all of the same sign and thus are allpositive in view of the asymptotic relation (29). The uniform positivity of n and n (and

thus existence of a positive d

such that 1/n = nn d

) follows immediately from thelemma below.

Lemma 4.2. For every d (0, /2) and r > 0 we have

sup(,)

supnZ+

log |S(2n)/2n| < , sup(,)

supnZ+

log |C(2n)| < ,

where S and C are constructed via (13) from the sequences and, and the suprema are

taken over (,) N(d, r).Proof. We assume first that n

= 0. By (13), we have

S(2n)/2n = 222n

2n2

kN, k=n

22k 22n2k2

.

Dividing both sides by

cos n =d sin z

dz

z=n

= 2

kN, k=n

k2 n2k2

,

we conclude that

|S(2n)/2n| =2

2n2n2

kN, k=n

2

2k 2

2n2(k2 n2);

for n = 0 the direct calculations give

lim0

|S()/| = 2kN

22k2k2

.

Recall that 2k := 2k k and set

an,k :=2n 2k(n

k)

,

with a0,0 = 1 and an,n = 0 if n N; then22k 22n

2(k2 n2) =

1 + an,k

1 + an,k

and2

|S(2n)/2n| =kZ

(1 + an,k).

In what follows, all summations and multiplications over the index set Z will be taken in the principal value

sense and the symbol V.p. will be omitted.

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50 Hryniv R.O.

Since the sequence (n) is 2d-separated for every (,) N(d, r), we have 1 + ak,n 2d/for all n Z+ and all k Z. Therefore, with

K := maxx1+2d/

log(1 + x) xx2

< ,

we get the estimate logkZ

(1 + an,k)

kZ

an,k + K

kZ

a2n,k, (32)

provided the two series converge.

Clearly, k=n

1

n k = 0,

and thus kZ

an,k

=

1

k=n

2kk n

r

3

by the CauchyBunyakovskiSchwarz inequality (recall that

kZ 22k r2 by the definitionof the set N(d, r) and

k=n(k n)2 = 2/3). Next, the inequality

a2n,k 222k

2(k n)2 +222n

2(k n)2for k = n yields

kZ

a2n,k 4r2k=n

1

2(k n)2 =4r2

3.

It follows from (32) that logkZ

(1 + an,k) (3r + 4Kr2)/3,

where the constant K only depends on d.

Similarly, we find that

|C(2n)| =

kZ+

22k+1 22n2(k + 1

2)2

= kZ+

22k+1 22n2(k + 1

2) 2n2

and then mimic the above reasoning to establish the other uniform bound. The lemma is

proved.

Proof of Theorem 3. Combining the results of Lemmata 4.1 and 4.2, we conclude that the

mappings (26) enjoy all the properties stated in Theorem 3, and thus so does the mapping

(,) 1. In virtue of Proposition 4.1 this completes the proof of the theorem.Proof of Theorem 1. Analyticity and Lipschitz continuity on bounded sets of the inverse

spectral mapping

N L2(0, 1)is the direct consequence of those for the mappings (25) and (15) established in Theorems 3

and 2 respectively.

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5 Some extensions

The results proved above for the class of impedance SturmLiouville operators with real-

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