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

Numerical index of vector-valued function spaces

Miguel Martı́n, Rafael Payá · Studia Mathematica · 2000

We show that the numerical index of a $c_0$-, $l_1$-, or $l_∞$-sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and $L_1(μ,X)$ (K any compact Hausdorff space, μ any positive measure) h

Cited by 61Open access

Conditional weak compactness in vector-valued function spaces

Marian Nowak · Proceedings of the American Mathematical Society · 2001

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an ideal of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 0"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -finite measure space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma normal upper Sigma comma mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Omega ,\Sigma ,\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E prime"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">E^{\prime }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Köthe dual of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s u p p upper E Superscript prime Baseline equals normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mi>supp</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal"> ′ </mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {supp} E^{\prime }=\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma double-vertical-bar dot double-vertical-bar Subscript upper X Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mo> ⋅ </mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,\|\cdot \|_{X})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real Banach space, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">X^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the topological dual of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a subspace of the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 0 Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi>

Cited by 3Open access

Extreme contractions on continuous vector-valued function spaces

Hasan Al-Halees, Richard J. Fleming · Proceedings of the American Mathematical Society · 2006

An old question asks whether extreme contractions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are necessarily <italic>nice</italic> ; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper K comma upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(K,E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself is a Banach space. We show that every extreme contraction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper K comma upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(K,E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to itself which maps extreme points to elements of norm one is nice, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sequence space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">c_{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Cited by 6Open access

Jointly separating maps between vector-valued function spaces

Ziba Pourghobadi, Masoumeh Najafi Tavani, Fereshteh Sady · arXiv (Cornell University) · 2018

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex Banach spaces, and $A(X,E)$ be a subspace of $C(X,E)$. In this paper we study linear operators $S,T: A(X,E) \lo C(Y,F)$ which are jointly separating, in the sense that $\coz(f) \cap \coz(g) = \emptyset$ implies that $\coz(Tf) \cap \coz(Sg)=\emptyset$. Here $\coz(\cdot)$ denotes the cozero set of a function. We characterize the general form of such maps between certain class of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied for a pair $T:A(X) \lo A(X)$ and $S:A(X,E) \lo A(X,E)$ of linear operators, where $A(X)$ is a regular Banach function algebra on $X$, such that $f\cdot g=0$ implies $Tf \cdot Sg=0$, for $f\in A(X)$ and $g\in A(X,E)$. If $T$ and $S$ are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between $X$ and $Y$ and, furthermore, $T^{-1}$ and $S^{-1}$ are also jointly separating maps.

Cited by 0Open access

On the numerical index of vector-valued function spaces

Mohamed Amine Khamsiz, Asuman Güven Aksoyx, Communicated W. Watkins · 2006

Let X be a Banach space and a positive measure. In this article, we show that nðLpð,XÞÞ limm nðlmp ðXÞÞ, 1 p &amp;lt;1. Also, we investigate the positivity of the numerical index of lp-spaces.

Cited by 0

Vector valued double sequence spaces defined by Orlicz function

Binod Chandra Tripathy, Bipul Sarma · Mathematica Slovaca · 2009

Abstract In this article we introduce some vector valued double sequence spaces defined by Orlicz function. We study some of their properties like solidness, symmetricity, completeness etc. and prove some inclusion results.

Cited by 49Open access

VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.

1972

This paper is a continuation of [8] and [9]. We begin with a general situation. The results obtained will yield information about compact sets in v.f.s.’s. Let X and a family {Y a e A} be Hausdorff topological spaces and let X--. Y be continuous maps. We suppose that ($) (x) for each a implies z x. Define Y and:X- * Y by (x) {(x)}. Then is continuous and one to one. A set S _ is said to be projec$ivvly cornpac if (S) is compact for each a. A sequence (x) X is projectively convergent if a(z) is convergent for each a. Other terms are defined similarly. The proofs of the following propositions present no difficulties; the proof of Proposition 1.1 uses Tychonoff’s theorem and that of Proposition 1.3(2) uses the finite intersection property characterization of compactness (see [6]). PROPOSITION 1.1 A se $ S X is compact if and only if (1) S is projectively compact, and (2) every projectively convergent net in S is convergent to a point in S. PROPOSXTO 1.2. Suppose A is countable. Then a set S X is sequentially compact (respectively relatively sequentially compact) if and only if (1) S is projec$ively sequentially compact (respectively relatively sequentially compact), and (2) every projec$ively convergen sequence in S is convergent to a point in S (respectively convergent). PROPOSTO 1.3. Suppose A; is countable. Then&amp;quot; (1) If in Y, the compact (respectively relatively compact) sets are sequentially compact (respectively relatively sequentially compact), then he same is $rue in X. (2) If in Y the sequentially compact sets are compact, then the same is true in X. (3) If inY,, the countably compact sets are compact, then the same is true in X. (4) If in Y the countably compact (respectively relatively countably conpact) sets are sequentially compact (respectively relatively sequentially compact), then the same is true in X. We now make some applications of the above three propositions. THEORE 1.4. If E is rnerizable and S(E) is a v.f.s, with a topology finer than the weak topology induced from (E) then the compact, sequentially compact,

Cited by 0

Generalized vector valued double sequence space using modulus function

Anindita Basu, P. D. Srivastava · Tamkang Journal of Mathematics · 2007

In this paper, we introduce a generalized vector valued paranormed double sequence space $ F^{2}(E,p,f,s) $, using modulus function $ f $, where $ p=(p_{nk}) $ is a sequence of non-negative real numbers, $ s\geq 0 $ and the elements are chosen from a seminormed space $ (E, q_{E}) $. Results regarding completeness, normality, $ K_{2} $-space, co-ordinatewise convergence etc. are derived. Further, a study of multiplier sets, ideals, notion of statistical convergence and ($ p_{nk} $ )-Ces\'aro summability in the space $ F^{2}(E,p,f,s) $ is also made.

Cited by 2Open access