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

УДК 517.55

Boundary Version of the Morera Theorem for a Matrix Ball of the Second Type

Gulmirza Kh. Khudayberganov* Zokirbek K. Matyakubov^

National University of Uzbekistan Vuzgorodok, Tashkent, 100174 Uzbekistan

Received 24.05.2014, received in revised form 26.08.2014, accepted 15.09.2014 In this article we prove a boundary Morera theorem for a matrix ball of the second type.

Keywords: matrix ball, automorphism, Poisson kernel, Morera theorem.

In this article we consider a boundary version of the Morera theorem for a matrix ball of the second type. Our starting point is Nagel and Rudin's result (see [1]), which says that if f is a continuous function on the boundary of a ball in Cn and the integral

/ f (>(eiv, 0, . . . , 0)) eivhp = 0,

J 0

for all (holomorphic) automorphisms ^ of a ball, then the function f is holomorphically extends into the ball. For classical domains an analog of a boundary Morera theorem was obtained in [7].

Let C [m x m] be the space of [m x m]-matrices with complex elements. We denote by Cn [m x m] the Cartesian product of n copies of C [m x m] :

Cn [m x m] = C [m x m] x ..... x C [m x m] .

Set (see, for example [2])

BI = {Z e Cn [m x m] : I - (Z, Z) > 0} ,

where (Z, Z) = ZiZ* + Z2Z* + ... + Z„Zj* is a 'scalar' product, I is the identity matrix [m x m], Z* = Z'v is the adjoint and transposed matrix to Zv, v = 1, 2, ...,n. BI is called a matrix ball (of the first type). Here I — (Z, Z) > 0 means that the Hermite matrix I — (Z, Z) is positively defined, i.e. all eigen values are positive. The skeleton of BI is the set

XI = {Z e Cn [m x m] : (Z, Z) = I} .

The domain BII in spaces Cn [m x m]:

BII = {Z e Cn [m x m] : I — (Z, Z) > 0, Z'v = Zv, v = 1, 2, ...,n} , (1)

where I is, as usual, the identity matrix of order m, is called a matrix ball of the second type (see [3]).

The skeleton of this domain is the following manifold:

XII = {Z e Cn [m x m] : (Z, Z) = I, ZV = Zv, v = 1, 2,..., n} .

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

Lemma 1. The domain B// has the following properties:

1) Bu is bounded;

2) Bu is a complete circular domain;

3) Bu and its skeleton X// are invariant under unitary transformations.

Proof. 1. The definition of the domain implies that each diagonal element of the matrix (Z, Z} is positive and less than 1, and the sum of the squares of the modules of all elements in Zv, v = 1, ...,n, does not exceed m. This implies that the matrix ball of the second type is bounded.

2. If Z e B// and a e C, |a| < 1, then

I - (aZ, aZ} = I - |a|2 (Z, Z} = I(1 - |a|2) + |a|2 (I - (Z, Z}) > 0.

3. Invariance under unitary transformations means that if U is a unitary matrix of order m, then for Z e B// we have UZ e B// and ZU e B//. Indeed,

i - (uz, uz} = i - uziziu* - uz2z2u* -.....- uznznu* =

= i - U (Z1Z1 + Z2Z2 +.... + Z„zn) U* = i - U (Z,Z} U* = U(i -(Z,Z})U* > 0,

and

(ZU, ZU} = (Z, Z}.

The invariance of the skeleton is proved similarly. □

We consider normalized Lebesgue measures ^ in B// and a on the skeleton X//, i.e.

I d^(Z) = 1 and I da(Z) = 1.

JBtt JXTT

IB II JXn

We define the space H1 (B//) as follows: a function f belongs to H1 (B//) if it is holomorphic in B// and

sup / |f (rZ)| da(Z) < to.

0<r<1j Xii

We fix a point A0 e X// (A0 = (A°,..., A^)) and consider the following embedding of a unit disk A in the domain B//

{W e Cn [m x m] : = ^A^, |£| < 1, v =1,...,n, } . (2)

By this embedding the boundary T of the disk A transforms into the disk on X//. If ^ is an automorphism of the domain B//, then the set (2) under the action of this automorphism becomes some analytic disk with the boundary on X//.

Theorem 1. Let f be a continuous function on X//. If f satisfies

f fW£A°))d£ = 0 (3)

Jt

for all automorphisms ^ of the domain B//, then the function f has a holomorphic extension F in B// of the class C(B//).

Proof. On X// the subgroup of the automorphisms leaving 0 fixed acts transitively (see [3]). Since X// is invariant with respect to unitary transformations, the condition (3) is satisfied for any point A e X//.

First of all, we parametrize manifold X// as follows: for Z e X// we put Z = el°U, where 0 < 9 ^ 2n, and in the matrix U1 the element in the left top corner is positive. We denote the

manifold of such matrices by X +. This way we parametrize not the whole set Xn, but some smaller set, which differs from X// by a set of zero measure.

The normalized Lebesgue measure da can be written as (Lemma 8.4 in [2])

da = ) = ),

2n 2ni £

where £ = el°, and the measure a1 is positive on X +.

Multiplying equality (3) by da1 and integrating over X+, from (3) we obtain

i f (V>(Z ))<q da(Z) = 0, (4)

JXn

where z^q are components of vector Z = (Z1, Z2, . . . , Zn), p, q =1,...., m , v =1,...., n.

We consider the automorphism translating the point A = (A1,..., An) from B// into 0 (see [3]). It is defined up to a generalized unitary transformation.

Then we substitute the automorphism in (4) instead of ^ and change variables W =

^-1(Z). We get

f f (W )<qV (W )da(^A(W ))=0, (5)

JX ii

where ^Aq'v are components of the automorphism By Corollary 7.7 from [2] we have

da(^A(W)) = P(A, W)da(W),

where P (A, W) is an invariant Poisson kernel for the matrix ball B// of the second type. Then, from the condition (5) we have that

i f (W (W )P (A, W )da(W )=0 (6)

JXii

for all points A = (A1,..., An) from B// and all p, q = 1,...., m , v = 1,...., n.

Thus, taking into account the properties of the Poisson integral of continuous functions, Theorem 1 follows from the next assertion.

Theorem 2. If for f € L1(X//) the equality (6) holds for all automorphisms ^a of domain B//, then f is a radial boundary value of some function F £ H 1(B//).

Proof. The invariant Poisson kernel for a matrix ball of the second type has the form

(

(det(/ - A1A1 -.....- A„A„))

P (A, W) =

(m + 1) n

%An ) '

det(1 - A1W 1 -.....- AnWn)|(m+1)n

(det(/ - A1A1 -.....- AnAn))

( m+1) n

n An ) '

(m + 1)n (m + 1)n

(det(/ - A1TW1 -.....- AnWn)) (det(/ - W1A1 -.....- WnAn))

We write the elements of matrices A and W in the vector form:

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

11, ..., ®1mi ....; flm1, ..., amm• ""• ®11, •'., ®1mi •

• an an ) = (\

-n an ) = n\a1 \\ \\an \

1 ^rry, rry~, i \ 11 pq 11 } *'''} || pq |

W = (W1,.",Wn) = (w11,.",w1m; ''''• wm^."^1^; —• wnl,''',

j1m; .

...; 1, ...,wmm) = (||wpq || , ||wpq ,

Lere ¡ap^ll = ¡«Vp! , ||wpq H = ¡w^! , P, q =1,...,m, v =1, ...,n. We shall compute

^ dP(A, W)

aP

(7)

pq dav p,q=1 v=1 dapq

Denote

I - W1A1 - ... - W„a4„ = ||aSJ-|| (s, j = 1,..., m),

where

m n

X ^ X ^ V — V V V V V '1

asj = ¿j - wskajk, ajk = akj, wsk = ^ j = 1 m,

k=1V=1

and is the Kronecker symbol. Calculations show that

ff _V d det(i - W1A1 - ... - Wn An) = 4pq d«v =

q=1 v=1 d apq

= det(i - W1A1 - ... - WnAn) - det(i - W1A1 - ... - WnAn)[p,p],

where det(i - W1A1 - ... - WnAn)[p,p] denotes the cofactor of the element app in the matrix I - W1A1 - ... - WnAn. Then

mn

V

EE

_V ddet(i - W1 A1 - ... - Wn^4n)

pq dav

p,q=1 V=1 dapq

= mdet(i - W1A1 - ... - W^n) - ^det(i - W1A1 - ... - W^n) [p,p] .

p=1

Similarly

m \_v ddet(i - A1 A1 - ... - An An) _

EE'

1 1 dapq

p,q=1 V=1 pq

m

= m det(i - A1A1 - ... - An An) det(i - A1A1 - ... - An An) [p,p] .

p=1

Therefore, the expression (7) is equal to

m (m + 1)

-nP (A, W)

E det(i - W1A1 - ... - Wn^4n)[p,p] p=1

det(i - W1A1 - ... - WnAn)

E det(i - A1 A1 - ... - AnAn)[p,p] p=1

det(i(m) - A1A1 - ... - An An)

= m("2+1)np(a, w)[Sp (i - W1A1 - ... - WnAn)-1 - Sp(i - A1A1 - ... - AnAn)-1]. (8) Here Sp, as usual, is the matrix trace.

m

2

m

An automorphism of the domain B// has the form (see [3])

n

^A(W) = R-1 (/ - W1A1 - ... - W„a4J(Wv - Av) Rvk, k = 1,..., n,

V=1

where R is a block matrix satisfying the condition

R' (i - A1A1 - ... - An An) R = I.

If the condition (6) is satisfied for the components of the map ^A(W), the same condition is satisfied for the components of the map

n

^A(W) = (/ - A1A1 - ... - AnAn)-1 (/ - W1 A1 - ... - WnAn)-1 (Wv - Av),

V=1

since matrices R, (/ - A1a41 - ... - AnAn) are nonsingular and depend only on A. Then from (6) we get

I f (WVAqV(W)P(A, W)da(W)=0, (9)

JXii

where (W) are the components of the map (W), (p, q =1,..., m, v =1,..., n). Now we compute the sum

mn p,q= 1 v=1

It is obvious that this expression is equal to Sp (^a(W), A}, since

m n

E Eapq^ = Sp f(i - A1a41 - ... - AnAn)-1 (/ - W1A1 - ... - W^)- x

(wia4i + ... + w„a4„ - aia4i - ... - a„a4„)] =

Sp

(/ - aia4i - ... - a„a4„P (/ - wia4i - ... - W„A4np x x ((I - aia4i - ... - a„a4„) - (/ - W1A1 - ... - w„a4„))] = = Sp

Using this, we get from (9)

(I - W1A1 - ... - W„A4„) 1 - (I - A1A1 - ... - A„A4„) ^ , (10)

m n

EEap, dF^ = 0, (11)

p,q=1 v=1

where

F (A) = i f (W )P (A,W )da(W) (12)

«11

is the Poisson integral of the function f.

The function F(A) is real analytic in the domain B//. We expand F(A) in a Taylor series in a neighborhood of 0,

F (A) Caiß aaäß,

H,lßl>o

where a = (||am1|| ,..., ||aMn||) and ß = (Hßpq11|,..., ||ßpqn||), (p, q = 1,..., m) are matrices with nonnegative integer elements and

m n m n

|a| =E , = II ü'

p,q=1 v=1 p,q=1 v=1

X

Then (11) implies

EEapqdgA1= E lei= 0.

p,q=1 v=1 pq |а|,|в|

It follows that for |в| > 0 all coefficients Ca,e are equal to zero. So, the function F(A) is holomorphic in Бц and belongs to the class H 1(B11).

If f is continuous on Xn, then the function F belongs to C(Б//) and its boundary values on Xn concide with f. □

The proof of this theorem shows that it remains true if the conditions (3) and (6) are satisfied only for those automorphisms фА, for which the point A = (A1,..., An) lies in some open set V с Бц. Therefore the following statement is true.

Theorem 3. If a function f € L1(X11) satisfies the condition (6) for all points lying in some open set V с Бц and for all components of the automorphism 'Фа, then f is a radial boundary value for some function F £ H 1(Б11) on Хц.

References

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

[2] G.Khudayberganov, A.M.Kytmanov, B.Shaimkulov, Complex analysis in matrix domains, Krasnoyarsk, Siberian Federal University, 2011 (in Russian).

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

[4] S.Kosbergenov, On multidimensional boundary Morera's theorem for matrix ball, Izvestiya VUZov. Matematika, (2001), no. 4, 28-32 (in Russian).

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

[6] F.R.Gantmakher, The theory of matrices, Chelsea Publition Company, 1977.

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

Граничный вариант теоремы Морера для матричного шара второго типа

Гулмирза Х. Худайберганов Зокирбек М. Матайкубов

В этой статье доказывается граничная теорема Морера для матричного шара второго типа. Ключевые слова: матричный шар, автоморфизм, ядро Пуассона, теорема Морера.