On tensor products of nuclear operators in Banach spaces (original) (raw)

Factoring operators over Hilbert-Schmidt maps and vector measures

Indagationes Mathematicae, 2009

We study the structure of Banach spaces X determinedby the coincidenceof nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-spacevalued mappings, it is shown that the class of Hilbert-Schmidtspaces arises as a'solution set' of the equation involving nuclear maps and the ideal of operators factoring through Hilbert-Schmidtmaps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equalityofthis latter ideal with the ideal of2-summingmaps. We shall also make use ofthis occasion to give an alternativeproofofa famous theorem ofGrothendieckusing some well-known results from vector measure theory.

On the vector Fourier multipliers for compact groups

Open Journal of Mathematical Sciences

This paper studies some properties of the Fourier multiplier operators on a compact group when the underlying multiplication functions (the symbols) defined on the dual object take values in a Banach algebra. More precisely, boundedness properties for such Fourier multiplier operators for the space of Bochner strong integrable functions and for the (vector) p-Fourier spaces are investigated.

Fourier multipliers, symbols and nuclearity on compact manifolds

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence, given a compact manifold endowed with a positive measure, we introduce a notion of the operator's full symbol adapted to the Fourier analysis relative to a fixed elliptic operator. We give a description of Fourier multipliers, or of operators invariant relative to the elliptic operator. We apply these concepts to study Schatten classes of operators and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.

On the Fourier Transform On Semi-direct Products With Compact Factor

Let H and K be locally compact groups and let θ : H → Aut(K) be a continuous homo-morphism. We consider G θ = H θ K the semi-direct product of H and K with respect to θ when K is compact. We study the Fourier transform of Banach space valued functions on G θ . We prove among other results the analogues of the Schur orthogonality relations and the Plancherel theorem.

Factorization theorem for 1-summing operators

Czechoslovak mathematical journal, 2011

We study some classes of summing operators between spaces of integrable functions with respect to a vector measure in order to prove a factorization theorem for 1-summing operators between Banach spaces.

On the nuclear trace of Fourier integral operators

Revista Integración, 2019

In this paper we characterise ther-nuclearity of Fourier integraloperators on Lebesgue spaces. Fourier integral operators will be consideredinRn,the discrete groupZn,then-dimensional torus and symmetric spaces(compact homogeneous manifolds). We also give formulae forthe nucleartrace of these operators. Explicit examples will be given onZn,the torusTn,the special unitary group SU(2),and the projective complex planeCP2.Ourmain theorems will be applied to the characterization ofr-nuclear pseudo-differential operators defined by the Weyl quantization procedure.

Vector Measures of Bounded Semivariation and Associated Convolution Operators

Glasgow Mathematical Journal, 2010

Let G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L1(G), the space of integrable functions on G into L1(G) X, the injective tensor product of L1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.

On the nuclearity of integral operators

Positivity, 2009

Let X be a nonempty measurable subset of \(\mathbb{R}^m\) and consider the restriction of the usual Lebesgue measure σ of \(\mathbb{R}^m\) to X. Under the assumption that the intersection of X with every open ball of \(\mathbb{R}^m\) has positive measure, we find necessary and sufficient conditions on a L 2(X)-positive definite kernel \(K : X \times X \rightarrow \mathbb{C}\) in order that the associated integral operator \(\mathcal {K} : L^2(X) \rightarrow L^2(X)\) be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of \(L^2(X \times X)\) .