Abstract
The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in ℂn into the little weighted‐type space is calculated. Some applications of the formula are given.
Highlights
Introduction and PreliminariesCharacterizing the compactness of composition or weighted composition operators, their differences, and Toeplitz operators between Banach spaces of analytic functions has attracted attention of numerous authors, and there has been a great interest in the matter of calculating or estimating essential norms of operators, see, for example, 1–8
The little weighted-type space Hv0 B Hv0 consists of all f ∈ Hv∞ such that lim v z f z 0
For vα z 1 − |z|2 α, α > 0, the standard weighted-type spaces, which we denote by Hα∞ and Hα0, are obtained
Summary
Characterizing the compactness of composition or weighted composition operators, their differences, and Toeplitz operators between Banach spaces of analytic functions has attracted attention of numerous authors, and there has been a great interest in the matter of calculating or estimating essential norms of operators, see, for example, 1–8. The little weighted-type space Hv0 B Hv0 consists of all f ∈ Hv∞ such that lim v z f z 0. For vα z 1 − |z|2 α, α > 0, the standard weighted-type spaces, which we denote by Hα∞ and Hα0, are obtained These spaces appear in the study of growth conditions of analytic functions, see, for example, 10, 11. The little Bloch-type space Bμ,[0] B Bμ,[0] consists of all f ∈ Bμ such that lim μ z Rf z 0 Both are Banach spaces with the norm f Bμ |f 0 | bμ f. Many results on weighted-type spaces of analytic functions and on operators between them are given in terms of the so-called associated weights see 10 and in terms of the weights.
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