HOLOMORPHIC SUPERPOSITION OPERATORS BETWEEN BANACH FUNCTION SPACES

2013 ◽  
Vol 96 (2) ◽  
pp. 186-197 ◽  
Author(s):  
CHRISTOPHER BOYD ◽  
PILAR RUEDA
Keyword(s):  
Large Class ◽  
Holomorphic Functions ◽  
Function Spaces ◽  
Weighted Spaces ◽  
Banach Function Spaces ◽  
Infinite Dimensional ◽  
Spaces Of Holomorphic Functions ◽  
Infinite Dimensional Holomorphy ◽  
Superposition Operators ◽  
Alpha Beta

AbstractWe prove that for a large class of Banach function spaces continuity and holomorphy of superposition operators are equivalent and that bounded superposition operators are continuous. We also use techniques from infinite dimensional holomorphy to establish the boundedness of certain superposition operators. Finally, we apply our results to the study of superposition operators on weighted spaces of holomorphic functions and the $F(p, \alpha , \beta )$ spaces of Zhao. Some independent properties on these spaces are also obtained.

scholarly journals Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
P. Rueda ◽  
E. A. Sánchez Pérez
Keyword(s):  
Type Theorem ◽  
Classical Case ◽  
Function Spaces ◽  
Dual Systems ◽  
Banach Function Spaces ◽  
Infinite Dimensional ◽  
Local Compactness ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Vector Valued

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

scholarly journals Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

2019 ◽  
Vol 277 (12) ◽  
pp. 108282 ◽  
Author(s):  
Karlheinz Gröchenig ◽  
Antti Haimi ◽  
Joaquim Ortega-Cerdà ◽  
José Luis Romero
Keyword(s):  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Spaces Of Holomorphic Functions

Composition operators on weighted spaces of holomorphic functions on the upper half plane

2018 ◽  
Vol 122 (1) ◽  
pp. 141
Author(s):  
Wolfgang Lusky
Keyword(s):  
Half Plane ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Bounded Operator ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Weight Functions ◽  
Spaces Of Holomorphic Functions ◽  
Upper Half Plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

Compactness and Almost Periodicity of Multipliers

1983 ◽  
Vol 26 (1) ◽  
pp. 58-62 ◽  
Author(s):  
G. Crombez
Keyword(s):  
Large Class ◽  
Function Spaces ◽  
Almost Periodic Functions ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Weakly Compact ◽  
Banach Function Spaces ◽  
General Method

AbstractThe question as to the existence of nontrivial compact or weakly compact multipliers between spaces of functions on groups has been investigated for several years. Until now, however, no general method which is applicable to a large class of function spaces seems to be knownIn this paper we prove that the existence of nontrivial compact multipliers between Banach function spaces on which a group acts is related to the existence of nonzero almost periodic functions.

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