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A new characterization of Bergman-Schatten spaces and a duality result. (English) Zbl 1192.46016

The authors consider some problems on duality of spaces of analytic matrices of the Bergman and Bloch type and study some characterizations by means of Taylor coefficients. If \(A=(a_{jk})\) and \(B=(b_{jk})\) are matrices of the same size (finite or infinite) the Schur (or Hadamard) product is defined by \(A*B=(a_{jk}b_{jk})\) and the space of Schur multipliers between spaces of matrices \(X\) and \(Y\), \(M(X;Y)\), is defined to be the set of matrices \(M\) such that \(M*A\in Y\) for any \(A\in X\). The authors consider functions \(r\to A(r)\) which are matrix-valued and use the correspondence with functions defined on the unit disc \(f_A(re^{it})=\sum_{k\in \mathbb Z} A_k(r)e^{ikt}\) where \(A_k(r)\) stands for the \(kth\)-diagonal matrix. In the particular case where \(A_k(r)=A_k r^k\), \(k\in \mathbb Z\), for a given upper triangular matrix \(A\) the matrix is called analytic matrix.
The spaces of matrices considered in the paper are the Bergman-Schatten classes for \(1\leq p<\infty\), to be denoted \(L^p(D,\ell^2)\), given by \(r\to A(r)\) such that \(A(r)\) belongs to the the Schatten class \(C_p\) and \(\int_0^1\|A(r)\|^p_{C_p} dr<\infty\) (which with the classical notation of Bochner integrable functions corresponds to \(L^p([0,1), C_p)\)) and its subspace of analytic matrices \(\tilde L_a^p(D,\ell^2)\) where \(A(r)=A*C(r)\) for some upper triangular matrix \(A\) and \(C(r)\) the Toepliz matrix associated to the Cauchy kernel \(\frac{1}{1-r}\). The obvious modification for \(p=\infty\) where the authors denote separately the spaces \(L^\infty(D,\ell^2)\) and \(\tilde L^\infty(D,\ell^2)\) whenever the functions \(r\to A(r)\) are assumed to be either \(w^*\)-measurable or strong measurable with values in \(B(\ell^2)\). The Bloch space \({\mathcal B}(D,\ell^2)\) is defined to be the space of analytic matrices \(A(r)\) such that \(\sup_{0\leq r<1}(1-r^2) \|A'(r)\|_{B(\ell^2)} + \|A_0\|_{B(\ell^2)}<\infty\) where \(A'(r)=\sum_{k=0}^\infty A_kkr^{k-1}\). Also the little Bloch space \({\mathcal B}_0(D,\ell^2)\) is defined in the usual way. Motivated by the situation for Toeplitz matrices the Bergman projection is defined by
\[ P(A)_{ij}= 2(j-i+1)r^{j-i}\int_0^1 a_{ij}(s)s^{j-i+1} ds \]
for \(i\leq j\) and zero otherwise.
The results about the boundedness of the Bergman projection on \(L^p(D,\ell^2)\) for \(1<p<\infty\) proved by N. Popa [“Matricial Bloch and Bergman-Schatten spaces”, Rev. Roum. Math. Pures Appl. 52, No. 4, 459–478 (2007; Zbl 1174.46015)] are extended to the case \(p=1\) and \(p=\infty\) by showing its boundedness from \(L^\infty(D, \ell^2)\) into \({\mathcal B}(D,\ell^2)\) and the boundedness of the modified Bergman projection \(P_2\) on \(L^1(D,\ell^2)\). Those facts allow the authors to obtain the expected duality results between both spaces. They also deal with a different problem and get an extension of the results by M. Mateljevic and M. Pavlovic [“\(L^{p}\)-behaviour of the integral means of analytic functions”, Stud. Math. 77, No. 3, 219–237 (1984; Zbl 1188.30004)] to the matricial case getting the condition for a matrix to belong to \(L^p_a(D, \ell^2)\) in terms of a summability condition on the CesĂ ro means of the matrix, namely \(\sum_{n=0}^\infty \frac{1}{n+1}\|\sigma_n(A)\|_p^p<\infty.\)

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46B10 Duality and reflexivity in normed linear and Banach spaces
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