Composition Operators on Some Banach Spaces of Harmonic Mappings

Abstract

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

On Characterizations of Weighted Harmonic Bloch Mappings and Its Carleson Measure Criteria

“…In the last few decades, the Banach spaces of analytic functions on D have been gaining a great deal of attention, but for the harmonic extensions of analytic spaces, it is still limited. Besides [1] by F. Colonna, papers such as [2] for the study of the operator theory on some spaces of harmonic mappings, [3] for characterizations of Bloch-type spaces of harmonic mappings, [4] for composition operators on some Banach spaces of harmonic mappings, [5] for the study of harmonic Bloch and Besov spaces, [6] for the study harmonic Zygmund spaces, [7] for the study of harmonic ν-Bloch mappings and [8] for the study of harmonic Lipschitz-type spaces. For α > 0, α-Bloch space for harmonic mapping is defined such that…”

The Möbius Invariant QTH Spaces

“…In [18], the authors investigate the compactness and boundedness of C ϕ mapping into weighted Banach spaces of harmonic mappings. We also encourage the reader to see the additional references related to the harmonic mappings such as [ [21] [5], [16], [14], [15], [17], [13], [7], [8], [10], [11], [12], [17], [9]].…”

Composition operators on Harmonic Bloch-type spaces

“…Moreover, the set of all Harmonic Bloch functions, denoted by HB(1) or HB, forms a complex Banach space with the norm |||.||| HB (1) , given by…”

“…In each case one of the the main goals is to discover the connection between the properties of the inducing function ϕ and the operator theoretic properties of C ϕ , for example, being bounded, compact, invertible, normal, subnormal, isometric, closed range, Fredholm, and many others. Extensive references for many of the known results on the subject can be found in [1,2,6,10,12,13,14]. In [4,5] we characterized bounded, compact and fredholm composition operators on Harmonic Bloch function spaces.…”