Generalized Köthe p-dual spaces (original) (raw)

Duality theory for p-th power factorable operators and kernel operators

The present work is devoted to the analysis of a particular class of (linear and continuous) operators between Banach function spaces. The aim is to advance in the theory of the so-called p-th power factorable operators by analyzing all the aspects of the duality. This class of operators has proved to be useful both in the factorization theory of operators on Banach function spaces (Maurey-Rosenthal theory) and in Harmonic Analysis (optimal domains for the Fourier transform and convolution operators). In order to develop the corresponding duality theory and some applications, a new class of operators with extension properties involving both the operator and its adjoint is defined and studied. This is the family of the (p, q)-th power factorable operators, for 1 ≤ p, q < ∞, that can be characterized by means of a canonical factorization scheme through the p-th power space of the domain space and the dual of the q-th power space of the dual of the codomain space. An equivalent diagram factoring such an operator through L p (m) and L q (n) for suitable vector measures m and n is also obtained, and this becomes the main tool (Chapter 3 and Chapter 4). Some other preliminary results concerning p-th powers of Banach function spaces are also necessary for constructing the above mentioned ones (Chapter 2). Using these tools, some results characterizing the optimal range-the smallest Banach function space in which the operator can take values-for operators from a Banach space into a Banach function space are given (Chapter 3). Also, the idea of optimal factorization of an operator optimizing a previous one is developed and formally presented in terms of the diagram that a (p, q)-th power factorable operator must satisfy (Chapter 4). All these results extend the nowadays well known computation of the optimal domain for operators on Banach function spaces by means of vector measures. These computations have provided relevant results in several fields of the mathematical analysis by means of a description of the biggest Banach function spaces to which some special relevant operators-for instance, the Fourier transform and the Hardy operator-can be extended. The theory is applied for finding new results in some concrete fields: as interpolation theory for operators between Banach function spaces, kernel operators (Chapter 5

The generalized multiplier space and its Köthe-Toeplitz and null duals

Mathematical Communications, 2017

The purpose of the present study is to generalize the multiplier space for introducing the concepts of alphaB-, betaB-, gamaB-duals and NB-duals, where B = (b_{n,k}) is an infinite matrix with real entries. Moreover, these duals are computed for the sequence spaces X and X(\delta), where X\in{l_p; c; c_0} and 1< p<\infty.

Factorization theorems for multiplication operators on Banach function spaces

Let X Y and Z be Banach function spaces over a measure space (⌦, ⌃, µ). Consider the spaces of multiplication operators X Y 0 from X into the Köthe dual Y 0 of Y , and the spaces X Z and Z Y 0 defined in the same way. In this paper we introduce the notion of fac-torization norm as a norm on the product space X Z · Z Y 0 ✓ X Y 0 that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with di↵erent factorization norms. Lozanovskii, Reisner and Maurey-Rosenthal theorems are special cases of this general setting. In this paper we analyze the class d ⇤ p,Z of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.

Notes on the spaces of bilinear multipliers

A locally integrable function m(ξ, η) defined on R n × R n is said to be a bilinear multiplier on R n of type (p 1 , p 2 , p 3) if Bm(f, g)(x) = Z R n Z R nf (ξ)ĝ(η)m(ξ, η)e 2πi(ξ+η,x dξdη defines a bounded bilinear operator from L p 1 (R n) × L p 2 (R n) to L p 3 (R n). The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where m(ξ, η) = M (ξ − η) for a given M defined on R n is also addressed. R n f (x)e −2πi x,ξ dx. We shall use the notation M p,q (R n) (respect.M p,q (R n)), for 1 ≤ p, q ≤ ∞, for the space of distributions u ∈ S ′ (R n) such that u * φ ∈ L q (R n) for all φ ∈ L p (R n) (respect. for the space of bounded functions m such that T m defines a bounded operator from L p (R n) to L q (R n) where T m (φ)(ξ) = m(ξ)f (ξ).) We endow the spaceM p,q (R n) with the "norm" of the operator T m , that is m p,q = T m. Let us start off by mentioning some well known properties of the space of linear multipliers (see [1, 14]): M p,q (R n) = {0} whenever q < p, M p,q (R n) = M q ′ ,p ′ (R n) for 1 < p ≤ q < ∞ and for 1 ≤ p ≤ 2, M 1,1 (R n) ⊂ M p,p (R n) ⊂ M 2,2 (R n). We also have the identificationsM

Idempotent Multipliers on Spaces of Continuous Functions with p-Summable Fourier Transforms

Proceedings of the American Mathematical Society, 1980

Let G denote a compact abelian group, and Ap the space of functions continuous on G and having p-summable Fourier transforms. The idempotent multipliers from Ap to A q are characterised for;», q e [1, 2]. Throughout G will denote a compact Hausdorff abelian group with character group T. Given p G [1, 2] we shall write Ap for the Banach space of functions continuous on G with /j-summable Fourier transforms, and normed by ||/|| = U/H» + WfWp-Note that we can identify A2 with C and A ' with A, the spaces of functions on G that are continuous and have absolutely convergent Fourier series respectively. An application of the Hahn-Banach theorem shows that the dual (Ap)' of Ap can be identified with M + Flp, where M is the space of Radon measures on G and Flp is the space of pseudomeasures on G with Fourier transforms in lp'. Here p' is the exponent conjugate to p, that is, p' = p/(p-1) with the usual convention ifp = 1. The duality is expressed by h(f) = ix*/(0) + <x*/(0), / G Ap, where h G (Ap)', ¡x G M and o G FF'. We shall write (Ap, A9) for the set of pseudomeasures that are multipliers from Ap to A9; that is, o G (AP,A9) if and only if for every/ G A" there exists g G A9 with a */= g. Using the above characterisation of the dual of Ap the following result can be proved.

On factorization of operators through the spaces lp.l^p.lp.

We give conditions on a pair of Banach spaces X and Y, under which each operator from X to Y, whose second adjoint factors compactly through the space l p , 1 ≤ p ≤ +∞, itself compactly factors through l p. The conditions are as follows: either the space X * , or the space Y * * * possesses the Grothendieck approximation property. Leaving the corresponding question for parameters p > 1, p = 2, still open, we show that for p = 1 the conditions are essential.

Products and Factors of Banach Function Spaces

2009

Abstract. Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E·M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F. Introduction. Let (X, Σ, µ) be a complete σ-finite measure space. By L0(X, µ) we will denote the set of all measurable functions which are finite a.e.. As usual we will identify functions equal almost evrywhere. A linear subspace of L0(X, µ) is called a Köthe function space if it is normed space which is an order ideal in L0(X, µ), i.e,, if f ∈ E

Extrapolation theorems for (p,q)(p,q)(p,q) -factorable operators

Banach Journal of Mathematical Analysis, 2018

The operator ideal of (p, q)-factorable operators can be characterized as the class of operators that factors through the embedding L q (µ) → L p (µ) for a finite measure µ, where p, q ∈ [1, ∞) are such that 1/p + 1/q ≥ 1. We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through rth and sth power factorable operators, for suitable r, s ∈ [1, ∞). Thus, they also factor through a positive map L s (m 1) * → L r (m 2), where m 1 and m 2 are vector measures. We use the properties of the spaces of u-integrable functions with respect to a vector measure and the uth power factorable operators to obtain a characterization of (p, q)-factorable operators and conditions under which a (p, q)-factorable operator is r-summing for r ∈ [1, p]. p (see [5, Theorem 19.3]). Maurey [12] also studied the class of operators that factor through L p-spaces of a finite measure, providing an extrapolation theorem for p-summing operators