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Multipliers on vector valued Bergman spaces. (English) Zbl 1065.42009

This paper is concerned with vector-valued Bergman spaces \(B^{p}(X)\). Given a complex Banach space \(X\) and \(p\in [1,\infty)\), an analytic function \(f\) from the unit disk \(D\) into \(X\) is said to belong to \(B^{p}(X)\) if \(\| f(z)\| _{X}^{p}\) is an integrable function with respect to the Lebesgue area measure \(dA(z)\) on \(D\).
A sequence \((T_{n})_{n}\) of bounded linear operators between two Banach spaces \(X\) and \(Y\) is said to be a multiplier between \(B_p(X)\) and \(B_q(Y)\) if for any function \(f(z)=\sum_{n=0}^\infty x_n z^n\) in \(B_p(X)\), \(x_{n}\in X\), the function \(g(z)=\sum_{n=0}^\infty T_n(x_n) z^n\) belongs to \(B_q(Y)\). Multipliers between \(B_{p}(X)\) and the space \(l_{q}(Y)\), the space of \(q\)-summable sequences in \(Y\), are defined similarly. These multipliers, thus, generalize the concept of the coefficient multipliers which has been studied by dozens of authors since the works of Hardy, Littlewood, and Paley. Vector-valued multipliers have been studied by several authors over the past 10 years, the present authors included.
In the paper under review the authors study various conditions that are either necessary or sufficient for a sequence of operators in order to be a multiplier from \(B_p(X)\) into \(B_q(Y)\), or between \(B_p(X)\) and \(l_q(Y)\). They first give several preliminary results on the Taylor coefficients of functions in \(B_{p}(X)\), analogous to the ones given by A. Nakamura, F. Ohya and H. Watanabe [Proc. Fac. Sci. Tokai Univ. 15, 33–44 (1980; Zbl 0442.30032)] and by S. V. Shvedenko [Sov. Math., Dokl. 32, 118–121 (1985); translation from Dokl. Akad. Nauk SSSR 283, 325–328 (1985; Zbl 0603.32005)] in the scalar case. Next, they discuss the Fourier and Rademacher type and Bergman type and cotype of Banach spaces, properties that are used later in the paper. They also prove some vector-valued analogues of theorems for functions in the scalar Bloch space or Bergman spaces.
Among other results, the authors give a complete characterization of the multipliers from \(B_{1}(X)\) into \(l_{q}(Y)\) in terms of the standard mixed-norm spaces \(l(p,q,X)\) and a characterization (for the spaces \(X\) isomorphic to a Hilbert space) of all multipliers from \(B_{p}\) into \(l_{q}(Y)\), where \(2\leq p<\infty\) and \(1\leq q\leq\infty\), analogous to the known ones (see, for example, [M. Jevtić and M. Pavlović, Acta Sci. Math. 64, No. 3–4, 531–545 (1998; Zbl 0926.46022)]). Finally, they prove some vector-valued criteria for multipliers between vector-valued Bergman spaces.

MSC:

42A45 Multipliers in one variable harmonic analysis
46E40 Spaces of vector- and operator-valued functions
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
30H05 Spaces of bounded analytic functions of one complex variable
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