Weighted Composition Operators from Hardy to Zygmund Type Spaces

Abstract

This paper aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator from the Hardy spaces to the Zygmund type spaces Z = { ∈ ( ) : sup ∈ (1 − | | 2 ) | ( )| < ∞} and the little Zygmund type spaces Z ,0 in terms of function theoretic properties of the symbols and .

Essential Norms of Volterra Type Operators between Zygmund Type Spaces

“…(iv) Case α = 2. We have prove that (19) holds above. To prove (20), we consider another test function t a (z) = log 2 1−az .…”

“…Let u be a fixed analytic function on D and an analytic self-map ϕ : D → D. Define a linear operator uC ϕ on the space of analytic functions on D, called a weighted composition operator, by uC ϕ f = u•(f •ϕ), where f is an analytic function on D. It is interesting to provide a function theoretic characterization when ϕ and u induces a bounded or compact composition operator on various function spaces. Some results on the boundedness and compactness of concrete operators between some spaces of analytic functions one of which is of Zygmund-type can be found, e.g., in ( [1,4,5,6,11,12,13,14,18,19,21]).…”

Norm and Essential Norm of Composition Followed by Differentiation from Logarithmic Bloch Spaces toHμ∞

“…In [10], Stević calculated the norm of the operator from the classical Bloch space to ∞ . Recently there has been some interest in calculating operator norms and essential norms of composition and related operators (see, e.g., [11][12][13][14][15][16][17][18] and the references therein). Motivated by the papers [10,19], we continue here this line of research by calculating ‖ ‖ LB → ∞ .…”

Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

“…Liu and Yu in [18] studied the boundedness and compactness of the operator D between ∞ and Zygmund spaces. Ye and Zhou in [26] studied the boundedness and compactness of the weighted composition operators from Hardy to Zygmund type spaces. Stević in [27] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Blochtype space, the little Bloch-type space, the Zygmund space, and the little Zygmund space.…”