×

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 Cesàro 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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arazy, J., Some remarks on interpolation theorems and the boundedness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory, 1, 4, 453-495 (1978) · Zbl 0395.47030
[2] Arregui, J. L.; Blasco, O., Multipliers on vector valued Bergman spaces, Canad. J. Math., 54, 6, 1165-1186 (2002) · Zbl 1065.42009
[3] Arregui, J. L.; Blasco, O., Bergman and Bloch spaces of vector-valued functions, Math. Nachr., 261/262, 3-22 (2003) · Zbl 1044.46033
[4] Barza, S.; Persson, L. E.; Popa, N., A matriceal analogue of Fejer’s theory, Math. Nachr., 260, 14-20 (2003) · Zbl 1043.15020
[5] Bennett, G., Schur multipliers, Duke Math. J., 44, 603-639 (1977) · Zbl 0389.47015
[6] Blasco, O., On coefficient of vector-valued Bloch functions, Studia Math., 165, 2, 101-110 (2004) · Zbl 1067.46039
[7] Blasco, O., Introduction to vector valued Bergman spaces, (Function Spaces and Operator Theory. Function Spaces and Operator Theory, Univ. Joensuu Dept. Math. Rep. Ser., vol. 8 (2005)), 9-30 · Zbl 1100.46022
[8] Blasco, O., Operators on weighted Bergman spaces \((0 < p \leqslant 1)\) and applications, Duke Math. J., 66, 3, 443-467 (1992) · Zbl 0815.47035
[9] Blasco, O., Spaces of vector valued analytic functions and applications, (Geometry of Banach Spaces. Geometry of Banach Spaces, London Math. Soc. Lecture Note Ser., vol. 158 (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 33-48 · Zbl 0736.46024
[10] Edwards, R. E., Functional Analysis: Theory and Applications (1965), Holt/Rinehart/Winston: Holt/Rinehart/Winston New York · Zbl 0182.16101
[11] Marcoci, A. N.; Marcoci, L. G., A new class of linear operators on \(\ell^2\) and Schur multipliers for them, J. Funct. Spaces Appl., 5, 151-164 (2007) · Zbl 1146.47017
[12] Mateljevic, M.; Pavlovic, M., \(L^p\)-behaviour of the integral means of analytic functions, Studia Math., 77, 219-237 (1984) · Zbl 1188.30004
[13] Popa, N., Matriceal Bloch and Bergman-Schatten spaces, Rev. Roumaine Math. Pures Appl., 52, 459-478 (2007) · Zbl 1174.46015
[14] Shields, A. L., An analogue of a Hardy-Littlewood-Fejer inequality for upper triangular trace class operators, Math. Z., 182, 473-484 (1983) · Zbl 0495.47027
[15] Rudin, W., Real and Complex Analysis (1987), McGraw-Hill · Zbl 0925.00005
[16] Zhu, K., Operator Theory in Banach Function Spaces (1990), Marcel Dekker: Marcel Dekker New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.