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Compact subsets of spaces of holomorphic functions. (English) Zbl 1097.46504

Summary: M. Dörfler, H. Feichtinger and K. Gröchenig [Colloq.Math.94, No. 1, 37–50 (2002; Zbl 1017.46014)] investigated compactness criteria in function spaces for the case of general coorbit spaces. These methods cannot be easily adapted for the spaces \(F_m\) and the Bergman spaces \(B^2(\Omega)\). We will be able to derive a generalization of the above mentioned result to the spaces \(F_m\) and \(B^2(\Omega)\). Furthermore, we will be able to derive a sufficient compactness condition for subsets \(A\) of the Fock space in terms of the Taylor-expansion of the functions \(f\in A\).
We will introduce increasing norm-spaces, which are a natural generalization of the above mentioned spaces. The main resultfor the spaces \(F_m\) and \(B^2(\Omega)\) will follow from this result.

MSC:

46B50 Compactness in Banach (or normed) spaces

Citations:

Zbl 1017.46014
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