Disjointly strictly singular operators and interpolation

2007

Let L, M be Archimedean Riesz spaces and Lb(L, M) be the ordered vector space of all order bounded operators from L into M . We define a Lamperti Riesz subspace of Lb(L, M) to be an ordered vector subspace L of Lb(L, M) such that the elements of L preserve disjointness and any pair of operators in L has a supremum in Lb(L, M) that belongs to L. It turns out that the lattice operations in any Lamperti Riesz subspace L of Lb(L, M) are given pointwise, which leads to a generalization of the classic Radon-Nikodým theorem for Riesz homomorphisms. We then introduce the notion of maximal Lamperti Riesz subspace of Lb(L, M) as a generalization of orthomorphisms. In this regard, we show that any maximal Lamperti Riesz subspace of Lb(L, M) is a band of Lb(L, M), provided M is Dedekind complete. Also, we extend standard transferability theorems for orthomorphisms to maximal Lamperti Riesz subspace of Lb(L, M). Moreover, we give a complete description of maximal Lamperti Riesz subspaces on some...

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021

We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

1997

We study the possibility of obtaining the l∞-norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in Rn such that the l∞-norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the l∞-norm by any method which guarantees interpolation of not only linear operators ( = bilinear forms on Rn×Rn) but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the l∞-norm, or both have to be weighted l∞-norms. Introduction The problem discussed in this article was inspired by a question posed by N. Kalton (Workshop on Interpolation spaces, Haifa, 1990), which he formulated as follows: is l∞ a black hole? He noticed that it is impossible to obtain the space l∞ by the complex method construction starting from a wide range of Banach couples both non-isometric to l∞ (including, in particular, couples of Banach lattices). He asked wh...

arXiv: Functional Analysis, 2020

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-Muller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calderon function lattices. We also derive connections between the spectrum of interpolated operators.

Annals of Functional Analysis, 2018

We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if T is an order bounded disjointness-preserving operator, then T and |T | have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having b-AM-compact (resp., AM-compact) operators defined between them as having a modulus that is b-AM-compact (resp., AM-compact).

Arkiv för matematik, 1989

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

Journal of Mathematical Analysis and Applications, 2008

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators. introduction Read [Read91] presented an example of a strictly singular operator with no (closed non-zero proper) invariant subspaces. It remains an open question whether every positive strictly singular operator on a Banach lattice has an invariant subspace. The present paper contains several results in this direction.

Proceedings of the American Mathematical Society, 1997

We study the possibility of obtaining the l ∞ l_{\infty } -norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in R n {\mathbb {R}}^{n} such that the l ∞ l_{\infty } -norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the l ∞ l_{\infty } -norm by any method which guarantees interpolation of not only linear operators ( = bilinear forms on R n × R n ) {\mathbb {R}}^{n}\times {\mathbb {R}}^{n}) but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the l ∞ l_{\infty } -norm, or both have to be weighted l ∞ l_{\infty } -norms.

2000

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.

arXiv (Cornell University), 2022

In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattice and which brings together several classical classes of operators (as oweakly compact operators, b-weakly compact operators, M-weakly compact operators, L-weakly compact operators, almost Dunford-Pettis operators). As consequences, we give some new lattice approximation properties of these classes of operators.

Studia Mathematica, 2020

An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, S(X, µ), of all µ-measurable simple functions on σ-finite measure space (X, µ) into M (Y, ν), the class of ν-measurable functions on σ-finite measure space (Y, ν), and satisfies endpoint estimates of type: 1 < p < ∞, 1 ≤ r < ∞, λ ν ({y ∈ Y : |(T f)(y)| > λ})

AMERICAN MATHEMATICAL …, 2010

We study the class of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation results for strictly singular operators on Lp spaces. Given p < q, it is shown that an operator T bounded on Lp and Lq which is strictly singular on Lr for some r between p and q, then it is compact on Ls for every p < s < q.

Bulletin of the American Mathematical Society, 1974

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.

Proceedings of the American Mathematical Society, 1988

We apply the complex method of interpolation to families of infinite-dimensional, separable Hubert spaces, and obtain a detailed description of the structure of the interpolation spaces, by means of a unique "extremal" operator-valued, analytic function. We use these interpolation techniques to give a different proof, under weaker assumptions, of a theorem of A. Devinatz concerning the factorization of positive, infinite-rank, operator-valued functions.