Abstract: In  compactness criteria in function spaces are investigated for general coorbit spaces. These methods cannot be easily adopted 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 conditions 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, that are a natural generalization of the above mentioned spaces. The main result will be proven for the spaces $F_m$ and $B^2(\Omega)$ will follow from this result.
Keywords: Fock-space, Bergman-space, Increasing norm-space
Classification (MSC2000): 46B50
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