In Edmunds et al. [J Lond Math Soc 78(2):65-84, 2008], a representation of a compactlinear operator T acting between reflexive Banach spaces X and Y with strictly convex duals was established in terms of elements x(n...
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In Edmunds et al. [J Lond Math Soc 78(2):65-84, 2008], a representation of a compactlinear operator T acting between reflexive Banach spaces X and Y with strictly convex duals was established in terms of elements x(n) is an element of X projections P-n of X onto subspaces X-n which are such that boolean AND X-n = kerT and linear projections S-n satisfying S(n)x = Sigma(n-1)(j=1) xi(j)(x)x(j) where the coefficients xi(j)(x) are given explicitly. If kerT = compact and the condition (A): sup parallel to S-n parallel to < infinity is satisfied, the representation reduces to an analogue of the Schmidt representation of T when X and Y are Hilbert spaces, and also (x(n)) is a Schauder basis of X;thus condition (A) can not be satisfied if X does not have the approximation property. In this paper we investigate circumstances in which (A) does or does not hold, and analyse the implications.
Let s(n)(T) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by T f(x) = v(x) integral(x)(a) u(t) f(t)dt, x is an element of (a, b) (-infin...
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Let s(n)(T) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by T f(x) = v(x) integral(x)(a) u(t) f(t)dt, x is an element of (a, b) (-infinity < a < b < +infinity) and mapping a Banach function space E to itself. We investigate some geometrical properties of E for which C-1 integral(b)(a) u(x)v(x)dx <= lim(n ->infinity) inf ns(n) (T) <= lim(n ->infinity)sup ns(n)(T) <= C-2 integral(b)(a) u(x)v(x)dx under appropriate conditions on it and v. The constants C-1, C-2 > 0 depend only on the space E. (C) 2016 Elsevier Inc. All rights reserved.
In this paper we prove that, under certain conditions, Nicodemi extensions of compact multilinearoperators between Banach spaces are compact as well. An application of this result to the isometric/isomorphic theory o...
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In this paper we prove that, under certain conditions, Nicodemi extensions of compact multilinearoperators between Banach spaces are compact as well. An application of this result to the isometric/isomorphic theory of spaces of compact multilinearoperators is provided.
We survey some of the recent developments involving embeddings between function spaces. Emphasis is placed on improvements of classical Sobolev inequalities, the reduction of embedding questions to problems involving ...
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We survey some of the recent developments involving embeddings between function spaces. Emphasis is placed on improvements of classical Sobolev inequalities, the reduction of embedding questions to problems involving Hardy operators, and quantitative estimates of compactness of embeddings that have applications to the spectral theory of operators. We also consider a nonlinear eigenvalue problem which leads to a series representation of compact linear operators acting between Banach spaces, under mild restrictions on the spaces, thus establishing a complete analogue of E. Schmidt's classical Hilbert space theorem for compactoperators. Information about relevant embedding maps enables the Dirichlet problem for the p-Laplacian to be studied, and a brief discussion is given of the generalizations of the trigonometric functions that appear naturally in this connection.
A compact operator in a separable Hilbert space is of infinite order if it does not belong to any Schatten-von Neumann ideal. In the paper, upper and lower bounds for the regularized determinants of infinite order ope...
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A compact operator in a separable Hilbert space is of infinite order if it does not belong to any Schatten-von Neumann ideal. In the paper, upper and lower bounds for the regularized determinants of infinite order operators are derived. By these bounds. perturbations results for the regularized determinants are established. (c) 2008 Elsevier Inc. All rights reserved.
The degree of ill-posedness of a linear inverse problem is an important knowledge base to select appropriate regularization methods for the stable approximate solution of such a problem. In this paper, we consider ill...
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Abstract: Let $M$ be a real (or complex) Banach space and $C(Y)$ the space of continuous real (or complex) functions on the compact Hausdorff space $Y$. The unit ball of the space of bounded operators from $M$...
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Abstract: Let $M$ be a real (or complex) Banach space and $C(Y)$ the space of continuous real (or complex) functions on the compact Hausdorff space $Y$. The unit ball of the space of bounded operators from $M$ into $C(Y)$ is shown to be the weak operator (or equivalently, strong operator) closed convex hull of its extreme points, provided $Y$ is totally disconnected, or provided ${M^ \ast }$ is strictly convex. These assertions are corollaries to more general theorems, most of which have valid converses. In the case $M = C(X)$, similar results are obtained for the positive normalized operators. Analogous results are obtained for the unit ball of the space of compactoperators (this time in the operator norm topology) from $M$ into $C(Y)$.
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