scholarly journals Vector-valued holomorphic and harmonic functions

2016 ◽  
Vol 3 (1) ◽  
pp. 68-76
Author(s):  
Wolfgang Arendt
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Harmonic Functions ◽  
Dual Space ◽  
Holomorphic Functions ◽  
Bounded Functions ◽  
Vector Valued ◽  
Separating Subspace

AbstractHolomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

scholarly journals Abelian ergodic theorems for vector-valued functions

1975 ◽  
Vol 16 (1) ◽  
pp. 57-60 ◽  
Author(s):  
P. E. Kopp
Keyword(s):  
Banach Space ◽  
Ergodic Theorem ◽  
Convergence Theorem ◽  
Direct Application ◽  
Approximation Technique ◽  
Ergodic Theorems ◽  
Resolvent Equation ◽  
Continuous Parameter ◽  
Vector Valued Functions ◽  
Vector Valued

This note contains extensions of the Abelian ergodic theorems in [3] and [6] to functions which take their values in a Banach space. The results are based on an adaptation of Rota's maximal ergodic theorem for Abel limits [8]. Convergence theorems for continuous parameter semigroups are deduced by the approximation technique developed in [3], [6]. A direct application of the resolvent equation also enables us to deduce a convergence theorem for pseudo-resolvents.

scholarly journals SCALAR BOUNDEDNESS OF VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 325-333 ◽  
Author(s):  
MATÍAS RAJA ◽  
JOSÉ RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Sufficient Conditions ◽  
Vector Integration ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe provide sufficient conditions for a Banach space-valued function to be scalarly bounded, which do not require to test on the whole dual space. Some applications in vector integration are also given.

scholarly journals Weighted spaces of harmonic and holomorphic functions: sequence space representations and protective descriptions

1997 ◽  
Vol 40 (1) ◽  
pp. 41-62 ◽  
Author(s):  
Päivi Mattila ◽  
Eero Saksman ◽  
Jari Taskinen
Keyword(s):  
Unit Disk ◽  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Inductive Limits ◽  
Complemented Subspaces ◽  
Bounded Functions ◽  
Köthe Sequence Spaces

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

scholarly journals The Dunford-Pettis property on vector-valued continuous and bounded functions

1993 ◽  
Vol 48 (2) ◽  
pp. 303-311 ◽  
Author(s):  
Jose Aguayo ◽  
Jose Sanchez
Keyword(s):  
Banach Space ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Bounded Functions ◽  
Continuous Operators ◽  
Vector Valued

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

scholarly journals Weighted Vector-Valued Holomorphic Functions on Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Enrique Jordá
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Holomorphic Functions ◽  
Connected Subset ◽  
Weighted Banach Spaces ◽  
Vector Valued Functions ◽  
Vector Valued

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a functionfdefined in a subsetAof an open and connected subsetUof a Banach spaceX, with values in another Banach spaceE, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

Geometry of Spaces of Vector-Valued Harmonic Functions

1994 ◽  
Vol 46 (2) ◽  
pp. 274-283
Author(s):  
Patrick N. Dowling ◽  
Zhibao Hu ◽  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Harmonic Functions ◽  
Norm Convergence ◽  
Nikodym Property ◽  
Vector Valued

AbstractIt is shown that the space hp(D,X) has the Kadec-Klee property with respect to pointwise norm convergence in the Banach space X if and only if X has the Radon-Nikodym property and every point of the unit sphere of X is a denting point of the unit ball of X. In addition, it is shown that hp(D,X) is locally uniformly rotund if and only if X is locally uniformly rotund and has the Radon-Nikodym property.

scholarly journals Lipschitz measures and vector-valued Hardy spaces

2001 ◽  
Vol 25 (5) ◽  
pp. 345-356 ◽  
Author(s):  
Magali Folch-Gabayet ◽  
Martha Guzmán-Partida ◽  
Salvador Pérez-Esteva
Keyword(s):  
Banach Space ◽  
Hardy Spaces ◽  
Dual Space ◽  
Real Banach Space ◽  
Lipschitz Functions ◽  
Vector Valued

We define certain spaces of Banach-valued measures called Lipschitz measures. When the Banach space is a dual spaceX*, these spaces can be identified with the duals of the atomic vector-valued Hardy spacesHXp(ℝn),0<p<1. We also prove that all these measures have Lipschitz densities. This implies that for every real Banach spaceXand0<p<1, the dualHXp(ℝn)∗can be identified with a space of Lipschitz functions with values inX*.

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