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About 2,633 papers for “Vector-valued function”

OPERATOR-VALUED INTEGRAL OF A VECTOR-FUNCTION AND BASES

M. H. Faroughi · 2015

Abstract. In the present paper we are going to introduce an operator-valued inte-gral of a square modulus weakly integrable mappings the ranges of which are Hilbert spaces, as bounded operators. Then, we shall show that each operator-valued inte-grable mapping of the index set of an orthonormal basis of a Hilbert space H into H can be written as a multiple of a sum of three orthonormal bases. 1.

Cited by 1

Surjective isometries on absolutely continuous vector valued function spaces

Fernanda Botelho, James Jamison · Contemporary mathematics - American Mathematical Society · 2017

In this paper we give a representation for the surjective and linear isometries on spaces of vector valued absolutely continuous functions with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -integrable derivatives ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1&gt;p&gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). The range space is a Banach space which is smooth, reflexive and separable. Representations for the generalized bi-circular projections and the bounded hermitian operators are also derived.

Cited by 5

On the character space of Banach vector-valued function algebras

Mortaza Abtahi · arXiv (Cornell University) · 2016

Given a compact space $X$ and a commutative Banach algebra $A$, the character spaces of $A$-valued function algebras on $X$ are investigated. The class of natural $A$-valued function algebras, those whose characters can be described by means of characters of $A$ and point evaluation homomorphisms, is introduced and studied. For an admissible Banach $A$-valued function algebra $\mathcal{A}$ on $X$, conditions under which the character space $M(\mathcal{A})$ is homeomorphic to $M(\mathfrak{A}) \times M(A)$ are presented, where $\mathfrak{A}=C(X) \cap \mathcal{A}$ is the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. An illustration of the results is given by some examples.

Cited by 0Open access

Conditional and Relative Weak Compactness in Vector-Valued Function Spaces

Marian Nowak · 2005

Let E be an ideal of L o over a »-flnite measure space (“;˛;¶), and let (X;kikX) be a real Banach space. Let E(X) be a subspace of the space L o (X) of ¶-equivalence classes of all strongly ˛-measurable functions f : “ †! X and consisting of all those f 2 L o (X) for which the scalar function kf(i)kX belongs to E. Let E(X) ‚ stand for the order continuous dual of E(X). In this paper we characterize both conditionally »(E(X);I)-compact and relatively »(E(X);I)-sequentially compact subsets of E(X) whenever I is an ideal of E(X) ‚. As an application, we obtain a characterization of almost re‡exivity and re‡exivity of a Banach space X in terms of conditionally »(E(X);I)-compact and relatively »(E(X);I)-sequentially compact subsets of E(X).

Cited by 4

Composition operators on vector-valued analytic function spaces: a survey

Laitila, Jussi, Tylli, Hans-Olav · arXiv (Cornell University) · 2015

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

Cited by 0Open access

Characterizations of Disjointness preserving operators on vector-valued function spaces

Jyh-Shyang Jeang, Ying‐Fen Lin · Proceedings of the American Mathematical Society · 2007

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper C 0 left-parenthesis upper X comma upper E right-parenthesis right-arrow upper C 0 left-parenthesis upper Y comma upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T : C_0(X, E) \to C_0(Y, F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact (resp. completely continuous) if and only if <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper T f equals sigma-summation Underscript n Endscripts delta Subscript x Sub Subscript n Subscript Baseline circled-times h Subscript n Baseline left-parenthesis f right-parenthesis for all f element-of upper C 0 left-parenthesis upper X comma upper E right-parenthesis comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mi>n</mml:mi> </mml:munder> <mml:msub> <mml:mi> δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo> ⊗ </mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mtext>for all </mml:mtext> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{align*} Tf = \sum _n \delta _{x_n} \otimes h_n (f) \quad \text {for all } f \in C_0(X,E), \end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n Baseline colon upper Y right-arrow upper B left-parenthesis upper E comma upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h_n : Y \to B(E,F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n Baseline left-parenthesis y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h_n(y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact (resp. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is uniformly completely continuous).

Cited by 10Open access

UNIFORM APPROXIMATION OF VECTOR-VALUED FUNCTIONS USING A WEIGHT FUNCTION

Pai D. V. · Yokohama National University Repository (Yokohama National University) · 1972

In this paper, uniform linear approximation of functions with values in a real or complex Hilbert space is considered with respect to a generalized weight function. The existence, characterization, unicity and strong unicity of the best approximation and the continuity of the best approximation operator for this type of approximation problem are discussed.

Cited by 0Open access

Vector-Valued Multiplier Spaces and Summing Operators: AModulus Function Approach

Ramazan Kama, Fatma Bulak · Preprints.org · 2026

In this paper, we introduce and systematically investigate novel classes of vector-valued multiplier spaces associated with operator-valued series, utilizing the concepts of \( f \)-statistical and weak \( f \)-statistical convergence. We begin by studying the topological properties of these newly defined spaces, establishing that their completeness is completely characterized by the \( c_0(X)- \) multiplier convergence of the underlying series. Building upon this structural foundation, we then explore the precise relationships between these \( f \)-statistical spaces and classical statistical multiplier spaces, proving that they perfectly coincide under the assumption of a compatible modulus function. Furthermore, we define a natural summing operator acting on these spaces and conduct a detailed analysis of its mapping properties. By establishing necessary and sufficient conditions for the continuity and (weak) compactness of this summing operator, we obtain new characterizations for both \( c_0(X)- \) and \( \ell_\infty(X)- \) multiplier convergent series.

Cited by 0Open access

Some sharp function estimates for vector-valued multilinear integral operator

Xiaosha Zhou · Miskolc mathematical notes/Mathematical notes · 2013

In this paper, we establish some sharp estimates for certain vector-valued multilinear integral operators. The operators include Littlewood-Paley operators, Marcinkiewicz operators and the Bochner-Riesz operator. As an application, we obtain the .L p ; L q /-norm inequality for the vector-valued multilinear operators.

Cited by 1Open access

On l 1 subspaces of Orlicz vector-valued function spaces.

Bombal Gordón, Fernando · Library Open Repository (Universidad Complutense Madrid) · 1987

The author studies those Orlicz vector-valued function spaces that contain a copy or a complemented copy of l 1 . Precisely, given a finite complete measure space (S,Σ,μ) , a Young function Φ , and a Banach space E , let L Φ (S,Σ,μ,E) denote the vector space of all (classes of) strongly measurable functions f from S to E such that ∫Φ(k∥f∥)dμ&lt;∞ for some k&gt;0 , and let L Φ (μ)=L Φ (S,Σ,μ,R) . The author first extends a result of G. Pisier concerning vector-valued L p function spaces by showing that if l 1 embeds in L Φ (S,Σ,μ,E) , then l 1 embeds either in L Φ (μ) or in E . This result, combined with a result of E. Saab and the reviewer concerning the embedding of l 1 as a complemented subspace of the Banach space of all E -valued continuous functions on a compact Hausdorff space, is used to show that if in addition E is a Banach lattice, if Φ satisfies the Δ 2 -condition and if μ is nonpurely atomic, then L Φ (S,Ω,μ,E) contains a complemented copy of l 1 if and only if either L Φ (μ) or E contains a complemented copy of l 1

Cited by 0