CC BY
15
УДК 517.55
The Bergman and Cauchy-Szego Kernels for Matrix Ball of the Second Type
Gulmirza Kh. Khudayberganov* Uktam S. Rakhmonov^
National university of Uzbekistan Vuzgorodok, Tashkent, 100174, Uzbekistan
Received 10.05.2014, received in revised form 06.06.2014, accepted 19.06.2014 With the use of holomorphic automorphism of the matrix ball of the second type the validity of the integral Bergman and Cauchy-Sege formulae is proven in this article.
Keywords: matrix ball, Bergman kernel, Cauchy-Szego kernel, automorphism of the matrix ball.
10. Let us assume that C[m x m] is the space of complex matrices of size [m x m]. Direct multiplication of n matrices is denoted by Cn[m x m]. The set
B^l = {Z = (Zi,..., Zn) e Cn [m x m] : I(m) - (Z, Z) > o}
is referred to as matrix ball of the first type (see [6]). Here (Z, Z) = Z1ZJ + Z2 ZJ + ... + Zn Zn is the "dot" product, I is the unit matrix of size [m x m], ZJ = Z'v is the conjugate transpose of matrix Zv, v = 1,2,..., n, and I — (Z, Z) > 0 means that a Hermitian matrix is positive definite
that is all matrix eigenvalues are positive.
(2)
Matrix ball the second type Bm'n has the following form (see [7]): B^n = {Z = (Zi,..., Zn) e Cn [m x m] : I(m) — (Z, Z) > 0, Z^ = Z^, v =1,..., n} .
(2) (2)
Let us denote the Shilov boundary of a matrix ball Bm,n by Xm,n, that is,
Xi2),, = {Z e Cn [m x m] : (Z, Z) = I, ZV = Z^, v =1, 2,..., n} .
This domain was originally considered in [7] and a group of holomorphic automorphisms of
(2)
Bm,n was described. The purpose of this paper is to find kernels of the integral Bergman and Cauchy- Szego formulae in the matrix ball of the second type. The integral Bergman formula for the matrix ball of the first type has been found in [6 ].
20. Let us consider a point P = (P^P2, ...,Pn) e B^n. Mapping
n
Wk = R-1(/(m)- < Z,P >)-1 (Zs — Ps)Gsk,k = 1, ..,n, (1)
s = 1
that transforms point P into 0 is an automorphism of the matrix ball B^^n (see [7 ]). Here R is a matrix of size [m x m] and G is a block matrix of size [m x n]. They satisfy the following relations
R'(1(m)- < P, P >)R = I(m), G'(1(mn) - PJP)G = I(mn). (2)
*[email protected] [email protected] © Siberian Federal University. All rights reserved
Lemma 1. Real Jacobean JR of the mapping W = <^>p(Z) at the point Z = P is
(m + 1)(n+1) 2
T , ( det(1(m)_ <P,P>) .
jr9p = |-—-To •
V|det(/(m)_ < Z,P >)|°/
Proof. Let us find the real Jacobean JR of the mapping W = ^p(Z) at the point Z = P. It follows from (1) that
n n
dWfc = R-1(1(m)_ < Z,P >)-1^ dZjP*(/(m)_ < Z,P>)-1^(Zs - Ps)Gsfc +
i=1 s=1
n
+R-1(1(m)- <Z,P>)-1^dzsGsfc•
1
n
dWfc |z=P = R 1(1(m)- <Z,P>)-1£dZsGsfc.
s =1
/ G1fc \
dZ ® G = (dZ1; •••,dZn) . =I~n,
\ Gnk /
dW = R-1(1(m)_ <Z,P>)-1dZ <g> G^
Then we have
(P) = R-1(1(m)- < Z,P >)-1 << G,
where is the Jacobi matrix of the mapping ^>P. The sign < means the Kronecker product of two matrices. Taking into consideration properties of the Kronecker product (see [3]) and using relation (2), we obtain
' , m + 1 m+1 n
det (P) = (det R') 2 (det G') 2 n •
Then applying the result of Theorem 2.1.2 from (see [2, p.37]), we find the real Jacobean of the mapping ^Z. Since
' 2
jr¥z = det
then
m + 1 - , m+1 - , (m + 1)(n+1) ,
jr^z(Z) = det— (RR')det~ (GG') = det--2-(/(m)_ < Z,Z >)• (3)
Taking into account relations (2), we obtain
det(1 (m)_ < W,W >) = det(R-1(1 (m)_ <Z,P>)-1 )det(1 (m)_ <Z,Z>)x
x det((1 (m)_ < P, Z >)-1R'-1) =_-_det(1 (ml_ < Z' Z >)_=
VV ' ' det((1 (m)_ < Z,P >)R)det(R'(1 (m)_ <P,Z>))
det(1 (m)_ < Z, Z >)
det(1 (m)_ < Z, P >) det(1 (m)_ < P, Z >) det(RR')
I(m)_ < p,P >= R'-1R-1 = (RR')-1 det(1 (m)_ < P,P >) = det(RR')-1 _ det(RR') = det-1(1 (m)_ < P,P >)
det(1(m)- < Z,Z >)
det(1(m)- < Z,P >)(det(1(m)- < Z, P >))*(det(/(m)- < P, P >))-1
_ det(1(m)- < Z,Z >)det(1(m)- < P,P >) |det(/(m)- <Z,P>)|2 '
det(1 (»)- < W, W >) _ det(1 (m)- <P,P>)det(1 (m)-<Z,Z>).
(4)
|det(1< Z,P >)
Mapping = o o y-1 conserves 0. Therefore it is a generalized unitary mapping and the absolute value of the Jacobian determinant equals 1, i. e., = yW1 ◦ o . Then from relations (3) and (4) we obtain
(m + 1)(n+1) . TT, TT, s ^ , „ TT, TT, s x (m + 1)(n+1)
im + ijin+i) / , / ,
_ det-s-(/(m)_ < W, W >) _ /det(/(m)- < W, W >)
_ det(I(m)- <Z,Z>) Vdet(/(m)- < Z, Z >)
/ det(1(m)- <P,P>)
(m + i)(n+i) 2
|2 • (5)
V|det(/(m)_ < Z,P >)|2/
□
30. Let us consider the normalized Lebesgue measures v in the ball Bmi and a on the Shilov
boundary Xmli, i.e.
i 2 dv(Z) _ 1 and i da(Z) _ 1.
Following the procedure given in [6] for the Bergman kernel is defined as follows:
K(z,w) = (m+1)(n+1) —-, z g Bmn.
det-2-(/(m)_ <Z,W>)
In particular, when n =1, this kernel coincides with the Bergman kernel for the classical region of the second type (see [2]).
(2)
The Hilbert space of holomorphic functions in that are square integrable with respect
to Lebesgue measure dv is designated as H2(Bm,n), i.e., f G H2(Bm,n) if f is a holomorphic in fuction and
/(2) If(Z)|2 dv(Z) <
L2(XÍ2)„, d^) signifies the space of scalar functions f that are square integrable with respect
(2) (2) to the normalized Haar measure d^ on the Shilov boundary Xm,n of the matrix ball
Theorem 1. For each functionf G H 1(Bm)n) the following relation is true
f(Z) = / f(w)K(Z, w)dv(w), z G pm)„, w G xm)n.
Integral in this relation defines the orthogonal projection from space
L2(Bm2)n) to
space
h 2 (Bm2)n).
(2) (2) Proof. Let us consider a point P G Bm,n- Let us assume first that the function f GA(Bm,n)
is holomorphic function in B consider the following function
(f is holomorphic function in Bin,™ and it is continuous function on the closure B^n)- Let us
K (Z,P)
g(Z) = Kpp) f (Z )-
Then g GA(Bm2)„) and
f (P ) = g(P) = (g o ^-1)(0). (6)
Expanding f in a series of homogeneous polynomials and integrating it over the ball, we obtain
f (0)= / (2) f (W)dv(W).
/ d(2)
Taking into account this relation and relation (5) we have
f (B)=/ g(^-1(W ))dv(W). (7)
/ d(2)
After the change of variables ^-1(W) = U in (7), we obtain
f(P) = / (2) g(u)jrdv(u)= / (2) f(U)k(p,u)dv(U).
(2)
Due to the completeness of the matrix ball the space of functions A(Bm;„) is dense in the space H2(Bm,l). Then the theorem holds for functions f G L2(Bmi)„)- □
40. Let us build the Cauchy-Szego kernel for the matrix ball of the second type. We define the Cauchy-Szego kernel C(Z, W) as follows
C(z, w) = —(m+1)n *-, Z G Bm)„, w G Xm)„. (8)
det 2 (I(m) -(Z,W»
At n =1 the Cauchy-Szego formula coincides with the Cauchy-Szego kernel for the classical region of the second type [2].
This kernel is defined for all pairs (Z, W) G Cn[m x m] x Cn[m x m] such that the matrix
I(m) - (Z, W)
(2) (2)
is not degenerate matrix. In particular, the kernel is defined for Z G Bm;„, W G Xm;„.
The kernel C(Z, W) is a holomorphic function with respect to elements of the block matrix Z and it is a antiholomorphic function with respect to elements of the block matrix W. If f G L1(Bm>„) on Xm^ one can introduce the following integral
c[f](Z) = / (2) C(Z, W)f (W)da(W), Z G B^l, W G Xi^. (9)
(2)
Let us designate C[f] as Cauchy integral with respect to f- The operator that transforms f into C[f] we designate as Cauchy transformLemma 2. Cauchy transform commutes with the action of the unitary group namely,
C[f o Vu] = (C[f]) o ^u, f G L1(a).
Proof. Let us show that the following equality is true
C (Z,^-1W ) = C (^uZ,W). (10)
In fact, UU* = J(m), VV* = J(mn) are unitary and block unitary matrices. Then we have
C (z^W )
u (m+1)n . . , (m+1)n
det 2 (I(m) - (Z,^-1W)) det 2 (I(m) - (Z, U-1))
(m+1)n (m+1)n
det 2 (I(mn) - Z* • U*WV*) det 2 (yy* - yy*z • U* WV*)
1 1
= C (^Z,W ).
(m+1)n (m + 1)n
det 2 (J(mn) - (UZV)*W) det 2 (/(m) - (UZV, W)) Here we use the equality
det(J(m) - (Z, W)) = det(J(mn) - Z* • W),
which is true by virtue of Theorem 2.1.2 (see [2, p. 37]) for arbitrary Z = (Z1,...,Zn) and W = (W1,..., Wn). Since the measure a is invariant with respect to then
C[f o ^u] = / () C(Z,W)f (^uW)da(W) = / () C(Z,^-1 W)f (W)da(W) =
J X(2) / X(2)
= ( () C(^uZ, W)f (W)da(W) = (C[f]) o W
(2) 0 Xm,n
Theorem 2. For each functionf G H 1(Bi!)n) the following relation is true
f (Z) = y f (W)C(Z,W)da(W),Z G Bi2)«, W G X^,«. (11)
X.
m )n
Proof. Let us assume that f G H 1(ßi?,')n) and Z G ß^n- Let us express a point Z G Cn[mxm] as Z = ('Z, Zn), where 'Z = (Z1, •••, Zn-1)- By the lemma we can assume without loss of generality that Zn = 0, i.e. Z = ('Z, 0).
Let us introduce the following function
g(Z) = с(Z,Z)f (Z), Z G B2)n.
(2)
Because Zn =0 then the Cauchy-Szego kernel in Bmn coincides with the Bergman kernel
B(2) . Bm.n.
с (Z, Z ) = к (' z,' z ).
Further, for any W G Xm;n function g('W, Zn) is the holomorphic function with respect to Zn in the matrix circle 5
w„wn - ZnZn > o, (12)
•n"n SnSn
and it is continuous function in the closure of the circle. Therefore, it follows from [2, c. 91] that
g('W, 0) = / g('W,W„)da(W„), (13)
JSn
1
1
1
where Sn is the Shilov boundary of matrix disk (12) and dr(Wn) is the invariant Haar measure
(2)
on Sn. Let us integrate relation (13) over Bm;n-1.
According to Fubini's theorem, on the right-hand side we obtain
/X
(2)
) = C [f ](Z ).
Because g('W, 0) = K(Z,' W)f ('W, 0) then it follows from Theorem 1 that the integral on the left-hand side of (13) is f ('Z, 0) = f (Z).
The theorem is proved. □
References
[1] C.L.Siegel, Analytic functions of several complex variables, Lectures delivered at Institute for Advanced Study, Princeton, 1948-1949.
[2] Hua Lo-ken, Harmonic analysis of functions of several complex variables in the classical domains, Moscow, IL, 1959 (in Russian).
[3] P.Lankaster, The theory of matrices, Academic Press, New York-London, 1969.
[4] F.R.Gantmakher, The theory of matrices, Chelsea Publition Company, 1977.
[5] G.Khudayberganov, A.M.Kytmanov, B.Shaimkulov, Complex analysis in the matrix domains, Krasnoyarsk, Siberian Federal University, 2011 (in Russian).
[6] S.Kosbergenov, The kernel of Bergman matrix ball, Uzbek Mathematical Journal, (1998), no. 1, 42-49 (in Russian).
[7] G.Khudayberganov, B.B.Hidirov, U.S.Rakhmonov, Automorphisms of matrix balls, Acta NUUz, (2010), no. 3, 205-210 (in Russian).
Ядра Бергмана и Коши-Сеге для матричного шара второго типа
Гулмирза Х. Худайберганов Уктам С. Рахмонов
С помощью голоморфности автоморфизмов матричного шара второго типа доказана справедливость интегральных формул Бергмана и Коши-Сеге.
Ключевые слова: матричный шар, ядро Бергмана, ядро Коши-Сеге, автоморфизм матричного шара.
