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Zbl 1192.46016
Marcoci, L.G.; Persson, L.E.; Popa, I.; Popa, N.
A new characterization of Bergman-Schatten spaces and a duality result.
(English)
[J] J. Math. Anal. Appl. 360, No. 1, 67-80 (2009). ISSN 0022-247X

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 $1\le p<\infty$, 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 $\sup_{0\le r<1}(1-r^2) \|A'(r)\|_{B(\ell^2)} + \|A_0\|_{B(\ell^2)}<\infty$ 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 $i\le j$ and zero otherwise. The results about the boundedness of the Bergman projection on $L^p(D,\ell^2)$ for $1<p<\infty$ proved by {\it 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 {\it M. Mateljevic} and {\it 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 $\sum_{n=0}^\infty \frac{1}{n+1}\|\sigma_n(A)\|_p^p<\infty.$
[Oscar Blasco (Valencia)]
MSC 2000:
*46B28 Normed linear spaces of linear operators, etc.
46B10 Duality and reflexivity in normed spaces

Keywords: Schur multipliers; Bergman-Schatten class; Bloch spaces; duality

Citations: Zbl 1174.46015; Zbl 1188.30004

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