Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces

Abstract

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

New Characterizations for the Essential Norms of Generalized Weighted Composition Operators between Zygmund-Type Spaces

“…Therefore, essential norm estimates of bounded operators result in necessary and/or sufficient conditions for the compactness of such operators. Essential norm estimates of different types of operators between various classes of Banach spaces have been studied by many authors (see, for example, [2][3][4][5][6][7], and references therein).…”

“…Essential norms of generalized weighted composition operators in these cases have been studied in [2,3,6,15]. In [16], different characterizations for the boundedness and compactness of these operators between Bloch-type spaces are given, and also, their essential norms are investigated in [7].…”

“…The following lemma, which will be used in the next chapters, collects some useful estimates for the functions in Zygmund-type spaces (see, for example, [6], and references therein).…”

The product-type operators from logarithmic Bloch spaces to Zygmund-type spaces

“…Note that, for ψ 2 ≡ 0, we obtain the weighted differentiation composition operator. For some later results on the weighted differentiation composition operator on various spaces of analytic functions see, e.g., [11,23,31,51,[58][59][60].…”

New characterizations for the essential norms of generalized weighted composition operators between Zygmund type spaces

“…Essential norm estimates of different types of operators between various classes of Banach spaces have been studied by many authors. See, for example, [3,4,5,6,13,16] and references therein.…”