Boundary Morera theorem for the matrix ball of the third type Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2018, 11(1), 40—45

УДК 517.55

Boundary Morera Theorem for the Matrix Ball of the Third Type

Gulmirza Kh. Khudayberganov*

National University of Uzbekistan VUZ Gorodok, Tashkent, 100174 Uzbekistan

Bayram P. Otemuratov^

Karakalpak State University Nukus, 230112 Uzbekistan

Uktam S. Rakhmonov*

Tashkent State Technical University Tashkent, 100174 Uzbekistan

Received 18.01.2017, received in revised form 06.03.2017, accepted 06.08.2017 In the article we consider a boundary version of Morera's theorem for the matrix ball of the third type.

Keywords: matrix ball of the first type, matrix ball of the third type, Poisson kernel, Morera theorem. DOI: 10.17516/1997-1397-2018-11-1-40-45.

10. We have the following result of Nagel and Rudin [1] which says that if f is a continuous function on the boundary of the unit ball in Cn and the integral

/ f (^(eiv, 0,..., 0))eivdv = 0 0

for all (holomorphic) automorphisms ^ of the ball, the function f extends holomorphically into a ball. For the classical domains, the matrix ball of the first type, and the generalized upper half-plane the boundary analogs of the Morera theorem were obtained in [2-4].

20. Let Z = (Z1,... ,Zn) be a vector composed from square matrices Zj of order m over the field of complex numbers C. We can assume that Z is an element of the space Cm n. We introduce on this set of vectors a matrix 'scalar' product according to

(.Z,W) = ZiW + ••• + ZnW.■*,

where W* is a conjugate transpose of the matrix Wj. The set

B^n = {(Zi, ...,Zn)= Z E Cn [m x m] : I -(Z,Z) > 0} ,

is called a matrix ball (of the first type); here (Z, Z) = Z1Z* + Z2Z* + • • • + ZnZ* is the 'scalar' product, I is the identity [m x m]-matrix, Z* = Z'v is the conjugate transpose of Zv, v = 1,2,... ,n, [5]. Here I — (Z, Z) > 0 means that the Hermitian matrix I — (Z, Z) is positively defined, i.e. all its eigenvalues are positive.

* [email protected] [email protected] [email protected] © Siberian Federal University. All rights reserved

30. We consider a matrix ball B^n (of the third type) (see [6]: B^n = m,...,Zn)= Z e Cn [m x m]: I + (Z,Z) > 0, Z'v = -Zv , v = l,...,n] .

The skeleton (the Shilov boundary) of the matrix ball Bm,n is denoted by Xm,n, i.e.

X^n = {Z e Cn [m x m]: I + {Z, Z) =0, Z'v = -Zv, v = l,...,n] .

We fix a point A0 e X^n (A0 = (A0,..., A^) and consider the following embedding of a unit disc A in the domain Bm ^n

nJ

jW e Cm2n : Wj = tA0, j = 1,...,n, \t\ < l} . (1)

The boundary T of the disc A by this embedding is mapped into a circle lying on Xm,n. If 0 is an arbitrary (holomorphic) automorphism of B^n, then the set of the form (1) under the

(3

action of the automorphism goes into some analytic disc with boundary in Xm,n.

(3

Theorem 1. If the function f e C(Xm,n) satisfies the following equality

i f (0(tA0))dt = 0 (2)

jt

(3 (3

for all automorphisms 0 of the ball Bm,n, then f extends holomorphically in Bm,n to a function F of class C(B^n).

(3) (3) '

Proof. We parameterize the manifold Xm'n- For Z e Xm,n we put Z = eiVU where

0 ^ p ^ 2n, and the element u^ in the upper left corner of the Ui is a positive number. A manifold of such matrices is denoted by X +. Note that not the whole set X^n is parameterized in this way, but a set smaller th Lemma 1 ( [5]). The measure

(3

in this way, but a set smaller than Xm'n, differing by a set of measure 0.

da = h(U)dtda+(U), U e X +,

where h(U) is a smooth positive function, does not depend on t. Lemma 1 shows that the measure da can be written as

da = <d^dai(U) = ddai (U), 2n 2ni t

where t = e1^, the measure al is positive on X +.

Multiplying equation (2) by dal and integrating over X +, we obtain from (2)

>X.

(3)

f (0(Z ))z'ksda(Z ) = 0 (3)

where zlks are the components of the vector Z = (Z\,..., Zn), k,s = 1,... ,m, l = 1,... ,n.

(3 )

Consider an automorphism 0b3) that maps an arbitrary point A from Bm,n to 0 [6]. It is defined up to a generalized unitary transformation.

Then, substituting in the condition (3) instead of 0 an automorphism 0—l3) and making the

change of variables W = 0-(3) (Z), we obtain

I (3) f (W)0tSl(W)M0a(W)) = 0, (4)

j X3 n

where 0^ are the components of the automorphism 0b3) . □

Corollary 1 ( [7]). For any continuous function f defined on the skeleton Xm,n the Poisson transformation F = P[f ] is a real-analytic function in Bmn}n\Xn]n and continuous on B^n, and

F = f on X$n-

Corollary 1 shows that da(^A(W)) = P(A,W)da((W), where P(A,W) is an invariant Poisson kernel of the domain B^n-

Therefore, from the condition (3) we obtain

f (Wykl(W)P(A, W)da((W)) = 0

'X.

(3)

(5)

for all the points A from B^-n and for all k, s = 1,... ,m, l = 1,... ,n.

Thus, taking into account Corollary 2 on the properties of the Poisson integral [7] of continuous functions, Theorem 1 follows from the following assertion.

(3 )

Theorem 2. If the function f E C(Xm,n) and equation (5) holds for all automorphisms of

(3 ) (3 )

the domain Bm,n transforming a point A from Bm,n to 0 and for all k,s = 1,..., m, l = 1,... ,n,

(3

then the function is the radial boundary value of some function F E o-(Bin'n)-

(3

Proof. The invariant Poisson kernel for the domain Bm,n has the following form (see [7]) for

even m

P (A,W )

(det(I(m) + (A, A) )) (det(I(m) + (A, W)))2

(m — 1) n

(det(I(m) + A1A1 + ••• + AnAn)) (det(I(m) + A1W1 + ••• + AnWn))\

(m—1)n

(det(I(m) + A1A1 + ••• + AnAn))

(m—1)n

( ) (m—1)n ( ) (m—1)n

(det(I(m) + A1W1 + ••• + AnWn)) (det(I(m) + W1A1 + ••• + WriAri))

and for odd m

P (A,W )

(det(I(m) + (A, A)))

(det(I(m) + (A, W)))2

( det(I(m) + A1A1 + ••• + AnAn)) (det(I(m) + A1W1 + ••• + AnWn))2

(det(I(m) + A1A1 + ••• + AnAn)Y

(det(I(m) + A1W1 + ••• + AnWn)) 2 (det(I(m) + WA + ••• + WriAri

Let

A =(A1,...,An) =

0,

11

= 0, ^12,..., 01m ;om 1,..., a

1m;am1, . . ., am(m-1), 0; . . . ; 0, a12

0;... ; 0, 0^2, ■■

n n n

• 0'1m; ... ; am1, . . . , 0m(m-1), 0) =

\asp, . . . , anp I

W = (W1,...,Wn) =

= (0,wn,...,wL;wL1,...,w1

wm(m-l), 0;...

, Wnm; ... ; wlm 1, . . . , W^n(m-1), 0)

Iw1p,...,wnpl

where 11olPN = II— 01

sp ps sp

We find the expression

|—wPs|| , (s,p = l,...,m), l = l,...,n.

dP(A, W)

nn

do1P

s,p=1 1=1 sp

2

2

2

2

1

Denote

I(m) + WiAi + ■■■ + WnAn = \\o-qj II (q,j = 1,...,m),

where aqj = Sqj + J2 m=l E n=l Wqk _jk , _sp = -alps, WSp = -W'ps, q, j =1,...,m and Sqj is the

Kronecker delta.

Using the usual rule for differentiating a determinant for any s=1,... ,m we obtain

ddet(I(m) + WxAi + ■■■ + WTiATi) _sp x-l

m n

-l

p=l 1=1 d_sp

= det(I(m) + Wih + ■■■ + WnAn) - det(I(m) + W^i + ■ ■ ■ + WnAn)[s, s],

where det(I(m) + W1A1 + ■ ■ ■ + WnAn)[s, s] denotes the algebraic complement to the element as

in the matrix det(I(m) + W1A1 +-----+ WnAn).

Then

m ^_ ddet(I(m) + WiAi + ■■■ + WnAn) = 1 fl _Sp d_lsv

s,p=i l-i sp

m

= m det(I (m)+WiAi + ■■■ + WriAri) -J^ det(I(m) + WiAi + ■■■ + WnAn)[s, s].

s=i

Similarly,

m n_ ddet(I(m) + AiAi + ■■■ + AnAri) =

i T1 _sp d_Sp

s,p=i l-i sp

m

= m det(I (m)+AiAi + ■ ■ ■ + AriAri) -J^ det(I(m) + Ai Ai + ■■■ + AnAn)[s, s].

s=i

Hence for even m we have the following equality

m (m - 1) nP(A, W)

Em=i det(I + WiAi + ■■■ + WnAn)[s, s]

=i_

det(I + WiAi + ■■■ + WnAn)

m ;s=i

Zm=i det(I + AiAi + ■ ■ ■ + AnAn)[s, s]

det(I + AiAi + ■■■ + AnAn) = m {m~ 1) n p (a, w )[Sp(I + WiAi + ■■■ + WnAn— - Sp(I + AiAi + ■ ■ ■ + AriAriri]. (7) for odd m the following equality

m (m + 1) nP(A, W)

Zm=i det(I + WiAi + ■■■ + WnAn)[s, s]

det(I + WiAi + ■■■ + WnAn)

;s=i

Em=i det(I + AA + ■■■ + AnAn)[s, s]

det(I + AiAi + ■■■ + AnAn) = m (m+1) n p (a, w )[Sp(I + WiAi + ■ ■ ■ + WnAn)-i - Sp(I + AiAi + ■■■ + An Ari)-i]. (7')

Here SpW denotes the trace of W. The mapping of the form [6]

n

0a (W) = Q-i ((I + WiAi + ■■■ + Wn An))-iJ2 (Ws - As) Qsk, k = 1,...,n,

s=i

transforming a point A to the origin, is an automorphism of the matrix ball B^n, where Q is

the block matrix Q(I + A1A1 +-----+ AnAn)Q' = I.

If the condition (5) holds for the components of the map ta (W), the same condition holds for the components of the map

— 1 — 1 n

tA (W ) = (I(m) + A1A1 + ••• + AnAn) (I(m) + W1A1 + ••• + WnAn) Y,(Wa — As) ,

s = 1

because the matrices Q and I(m) + W1A1 + • • • + WnAn are non-degenerate and depend only on A.

Then from (5) we get

I (3) f (W(W)P(A, W)da((W)) = 0, where tAs!(W) are components of the map tA (W), (s,p = 1,..., m), v = 1,... ,n. Consider

the sum

m n

V

spv sp

s,p=l l = l

> ,p=i 1=1

Obviously, this expression is equal to Sp (фА (W) ,A), as [8-10]

m n

J2 J2alsp^Apl = Sp [(I(m) + A1A1 + ■ ■■ + AnAn)-1(I(m) + W1A1 + ■■■ + WnAn)-

s,p=l l=l

x {A1A1 + ■■■ + AnAn - W1A1-----WnAn)] =

(m) + A -, A +_____u A.. A. )-1(T(m) + W A +_____i- W.. A-)-1-

Sp

Sp

(I(m) + A1A1 + ■■■ + AnAn)-1 (I(m) + W1A1 + ■■■ + WnAn)- -x < ((I + A1A1 + ■■■ + AnAn) - (I + W1A1 + ■■■ + WnAn))] = (I(m) + W1A1 + ■■■ + WnAn)-1 - (I(m) + A1A1 + ■■■ + AnAn)-1

(8)

Comparing formulas (7) and (8), from the hypothesis of the theorem we obtain

t tdF^=0- <9)

s,p=11=1 db

sp

where F (A) = f (W )P (A,W )da(W) is the Poisson integral of the function f. □

j x(3)n

40. The proof of this theorem shows that it remains valid if the condition (5) holds only for

(3)

automorphisms tB(3 , for which the point A lies in an open set V C Bm,n. As Theorem 1, Theorem 2 can be generalized.

(3)

Theorem 3. Let the function f E C(Xm'n) and the condition (2) holds for all automorphisms

(3)

t that transform the origin to a point of some open set V C Bm'n- Then f holomorphically

(3) (3)

extends in the domain Bm,n to a function F E ff(Bfn)n). 50. Let A^ be an analytic disc

A^ = {Z : Z = t(tA0), |t| < 1},

where A0 is a fixed point of the skeleton X^l, and t is an automorphism of the domain B^I.

(3)

Then the boundary T^ of the analytic disc lies on Xm'n, since the automorphism maps points of the skeleton to the points of the skeleton.

1

x

From the Morera theorems we obviously get a corollary on functions with one-dimensional holomorphic extension property along analytic discs.

(3)

Corollary 3. If the function f e C(Xm,n) exdends holomorphically (in t) in analytic discs A^ for all automorphisms 0 (or for all automorphisms 0 that transform the origin to a point of

(3) (3)

some fixed open set V C Bm,n), then the function f extends holomorphically in Bm.n.

References

[1] A.Nagel, W.Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J., 43(1976), no. 4, 841-865.

[2] S.Kosbergenov, A.M.Kytmanov, S.G.Myslivets, On a boundary Morera theorem for classical domains, Siberian Math. J., 40(1999), no. 3, 506-514.

[3] G.Khudayberganov, B.T.Kurbanov, The boundary Morera theorem for generalized upper half-plane, Uzbekskiy Mat. Zh., (2002), no. 1, 78-83 (in Russian).

[4] S.Kosbergenov, On a multidimensional boundary Morera theorem for the matrix ball, Russian Math. (Iz. VUZ), 45(2001), no. 4, 26-30.

[5] G.Khudayberganov, A.M.Kytmanov, B.A. Shaimkulov, Complex analysis in the matrix domains, Sib. Federal Univ., Krasnoyarsk, 2011 (in Russian).

[6] G.Khudayberganov, B.B.Khidirov, U.S.Rakhmonov, Automorphisms of matrix balls, Dok-lady NUUz., (2010), no. 3, 205-210 (in Russian).

[7] U.S.Rakhmonov, Poisson kernel matrix ball for the third type, Uzbekskiy Mat. Zh., (2012), no. 3, 123-125 (in Russian).

[8] P.Lankaster, The theory of matrices, Academic Press, New York-London, 1969.

[9] F.R.Gantmacher, The theory of matrices, Chelsea Publishing Company, 1977.

[10] G.Khudayberganov, U.S.Rakhmonov, Z.Q.Matyakubov, Integral formulas for some matrix domains, Contemporary Mathematics , 662(2016), 89-95.

Граничная теорема Морера для матричного шара третьего типа

Гулмирза Х. Худайберганов

Национальный университет Узбекистана им. М. Улугбека ВУЗ городок, Ташкент, 100174, Узбекистан

Байрам П. Отемуратов

Каракалпакский государственный университет Ч.Абдирова, 1, Нукус, 230112, Узбекистан

Уктам С. Рахмонов

Ташкентский государственный технический университет

Ташкент, 100174, Узбекистан

В этой статье 'рассматривается граничный вариант теоремы Мореры для матричного шара третьего типа.

Ключевые слова: матричный шар первого типа, матричный шар третьего типа, ядро Пуассона, теорема Морера.