scholarly journals The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

2021 ◽  
Author(s):  
Karsten Kruse
Keyword(s):  
Real Axis ◽  
Hausdorff Space ◽  
Holomorphic Functions ◽  
Smooth Vector ◽  
Riemann Equation ◽  
Dependence Problem ◽  
Cauchy Riemann ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

AbstractThis paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

scholarly journals Multipliers of Modules of Continuous Vector-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liaqat Ali Khan ◽  
Saud M. Alsulami
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Normed Spaces ◽  
Locally Compact ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

In 1961, Wang showed that ifAis the commutativeC*-algebraC0(X)withXa locally compact Hausdorff space, thenM(C0(X))≅Cb(X). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing thatHomC0(X,A)(C0(X,E),C0(X,F))≃Cs,b(X,HomA(E,F)),whereEandFarep-normed spaces which are also essential isometric leftA-modules withAbeing a certain commutativeF-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

scholarly journals Extension of vector-valued functions and weak–strong principles for differentiable functions of finite order

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Karsten Kruse
Keyword(s):  
Finite Order ◽  
Hausdorff Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Differentiable Functions ◽  
Weighted Banach Space ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractIn this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field $$\mathbb {K}$$ K , which has weak extensions in a weighted Banach space $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) of scalar-valued functions on a set $$\Omega$$ Ω , to functions in a vector-valued counterpart $$\mathcal {F}\nu (\Omega ,E)$$ F ν ( Ω , E ) of $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) . Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.

scholarly journals Operators on locally convex spaces of vector-valued continuous functions

1987 ◽  
Vol 36 (2) ◽  
pp. 267-278
Author(s):  
A. García López
Keyword(s):  
Hausdorff Space ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Special Treatment ◽  
Locally Convex Spaces ◽  
Uniform Topology ◽  
Zero Sequence ◽  
Vector Valued ◽  
Locally Convex ◽  
Convex Spaces

Let E and F be locally convex spaces and let K be a compact Hausdorff space. C(K,E) is the space of all E-valued continuous functions defined on K, endowed with the uniform topology.Starting from the well-known fact that every linear continuous operator T from C(K,E) to F can be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.As a consequence we characterise the spaces E for which an operator T defined on C(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) in C(K,E).

scholarly journals The strict topology on a space of vector-valued functions

1979 ◽  
Vol 22 (1) ◽  
pp. 35-41 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Scalar Field ◽  
Topological Space ◽  
Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Strict Topology ◽  
Locally Compact ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

scholarly journals Mean value theorems and a Taylor theorem for vector valued functions

1981 ◽  
Vol 24 (1) ◽  
pp. 69-77
Author(s):  
Rudolf Výborný
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Extension Theorem ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

scholarly journals On approximation in weighted spaces of continuous vector-valued functions

1987 ◽  
Vol 29 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Regular Space ◽  
Local Convexity ◽  
Range Space ◽  
Completely Regular ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

scholarly journals Analyticity and quasi-Banach valued functions

1990 ◽  
Vol 42 (3) ◽  
pp. 369-382
Author(s):  
Antonio Bernal ◽  
Joan Cerdà
Keyword(s):  
Tensor Products ◽  
Locally Convex Spaces ◽  
Vector Valued Functions ◽  
Topological Tensor Products ◽  
Vector Valued ◽  
Locally Convex ◽  
Topological Tensor ◽  
Convex Spaces

We compare the definitions of analyticity of vector-valued functions and their connections with the topological tensor products of non-locally convex spaces.

scholarly journals The Arens Algebras of Vector-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
I. G. Ganiev ◽  
O. I. Egamberdiev
Keyword(s):  
Banach Algebra ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

A class of Arens algebras of vector Banach algebra valued functions is considered. It is shown that Arens algebra of vector-valued functions is complete locally convex metrizable algebra.

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