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Let Cp be the Schatten p-class of ℓ2 for 1

In this research paper, we develop some aspects of the theory of Fredholm linear operators acting in p-adic (or non-Archimedean) Banach spaces. In this regard, we establish sufficient conditions for the p-adic Fredholmness of the algebraic sum of unbounded linear operators. Next, we study the perturbation of p-adic upper semi-Fredholm operators under strictly singular operators. Moreover, we give some results concerning quasi-compact and p-adic lower semi-Fredholm operators. [ABSTRACT FROM AUTHOR]

We prove that any normalized block sequence in a Schreier space X ξ , of arbitrary order ξ < ω 1 , admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for dual spaces to Schreier spaces. Using these results, we examine the structure of strictly singular operators on Schreier spaces and show that there are 2 c many closed operator ideals on a Schreier space of any order, its dual and bidual space. [ABSTRACT FROM AUTHOR]

Our objective in this paper is to continue the investigation of the duality problem for the class of AWC operators. First, we introduce a new class of operators called semi weakly compact operators and we study the basic properties of this class. Also, we shall give some relations between almost weakly compact and semi weakly compact operators through the use of the adjoint. Finally, we will give a condition under which if the adjoint T ′ is AWC, then so is T. [ABSTRACT FROM AUTHOR]

We study the structure of strictly singular non-compact operators between Lp spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators T, for which the set of points (1/p, 1/q) ∈ (0, 1) × (0, 1) such that T:Lp →Lq is strictly singular but not compact contains a line segment in the triangle {(1/p, 1/q): 1 < p < q < 1 ∞} of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of Lp is also explored. [ABSTRACT FROM AUTHOR]

This paper is devoted to a characterization of the notion of generalized weakly demicompact linear operators acting on a Banach space introduced in [8]. Our results are formulated in terms of measures of non-strict-singularity and strictly singular operators. Theses results are used to discuss the incidence of some perturbation results on the behavior of the closure of unbounded 2 × 2 block operator matrices. [ABSTRACT FROM AUTHOR]

We refine some earlier results by Flores, Hernández, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator acting in X is compact. We show that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with the non-trivial Boyd indices has compact square, and that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space X whenever X ⊃ G , where G is the closure of L ∞ in the Orlicz space, generated by the function e u 2 - 1 . Moreover, a partial converse to the latter result is given under the assumption that X ⊂ L log 1 / 2 L . As an application we find rather sharp conditions, under which a Lorentz space Λ (2 , ψ) possesses the Kato property. In particular, Λ (2 , log - α (e / u)) , with 0 < α ≤ 1 , is a 2-DH and 2-convex rearrangement invariant space, which does not have the Kato property. This gives a negative answer to the question posed by Hernández, Semenov, and Tradacete. [ABSTRACT FROM AUTHOR]

Abstract Every infinite dimensional Banach space admits Riesz operators that are not finite rank. In this note we discuss conditions under which a Riesz operator, or some power thereof, is a finite rank operator. [ABSTRACT FROM AUTHOR]

In this paper, we study the strong convex compactness property on the hereditarily indecomposable Banach spaces denoted respectively by XGM and XAM constructed by T. Gowers and B. Maurey (1993) and Argyros-Motakis (2014). We prove in particular that the Bochner integral of a family of strictly singular operators is a strictly singular operator. Also, some properties of C0-semigroups defined on these Banach spaces are given. [ABSTRACT FROM AUTHOR]

In this work we introduce the disjointly improjective operators between Banach lattices. We investigate this class of operators. Also, we extend the Flores–Hernández’s theorem on the domination problem by disjoint strictly singular operator. [ABSTRACT FROM AUTHOR]

We study the interpolation and extrapolation properties of strictly singular operators between different L p spaces. To this end, the structure of strictly singular non-compact operators between L p − L q spaces is analyzed. Among other things, we clarify the relation between strict singularity and the L -characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators. [ABSTRACT FROM AUTHOR]

We construct a Banach space X of Gowers-Maurey type such that the algebra of bounded operators L (X) is a direct sum of an infinite dimensional reflexive Banach space and the operator ideal of strictly singular operators SS (X). [ABSTRACT FROM AUTHOR]

In this paper we are concerned with developing generalizing concepts of Dunford-Pettis operators analogous to the generalization of compact operators by strictly singular operators. Also, we give some new results concerning the domination problem in the setting of positive operators between Banach lattices. [ABSTRACT FROM AUTHOR]

In this paper we give several new results concerning domination problem in the setting of positive operators between Banach lattices. Mainly, it is proved that every positive operator $$R$$ on a Banach lattice $$E$$ dominated by an almost weakly compact operator $$T$$ satisfies that the $$R^2$$ is almost weakly compact. Domination by strictly singular operators is also considered. Moreover, we present some interesting connections between strictly singular, disjointly strictly singular and almost weakly compact operators. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the class of strictly singular operators and the class of finitely strictly singular operators are stable under perturbed domination. We prove a diagonal theorem, which is a vector valued version of the result due to V. Milman stating that the identity from to is finitely strictly singular when . Some applications are given. The paper is illustrated with examples in harmonic analysis and function theory: we prove that the Fourier transform from into is finitely strictly singular if and only if . In the case , some examples of different kind of behavior involving thin sets are given. Another result deals with the identity from the Hardy space to the Bergman space : it is finitely strictly singular for every . This result requires some new special inequalities between the norms on the Hardy and the Bergman spaces. [Copyright &y& Elsevier]

We prove that if Ф is a centralizer on Lp, where 0 < p < ∞, then there is a copy of ⨏2 inside Lp where Ф is bounded. If Ф is symmetric, then it is also bounded on a copy of ⨏q, provided 0 < p < q < 2. This shows that for a wide class of exact sequences 0 → Lp → Z → Lp → 0 the quotient map is not strictly singular, which generalizes a recent result of Jesús Suárez. [ABSTRACT FROM AUTHOR]

We prove that if Ф is a centralizer on Lp, where 0 < p < ∞, then there is a copy of ⨏2 inside Lp where Ф is bounded. If Ф is symmetric, then it is also bounded on a copy of ⨏q, provided 0 < p < q < 2. This shows that for a wide class of exact sequences 0 → Lp → Z → Lp → 0 the quotient map is not strictly singular, which generalizes a recent result of Jesús Suárez. [ABSTRACT FROM AUTHOR]

Let $$X(\mu )$$ be a p-convex ( $$1\le p<\infty $$) order continuous Banach function space over a positive finite measure $$\mu $$. We characterize the subspaces of $$X(\mu )$$ which can be found simultaneously in $$X(\mu )$$ and a suitable $$L^1(\eta )$$ space, where $$\eta $$ is a positive finite measure related to the representation of $$X(\mu )$$ as an $$L^p(m)$$ space of a vector measure $$m$$. We provide in this way new tools to analyze the strict singularity of the inclusion of $$X(\mu )$$ in such an $$L^1$$ space. No rearrangement invariant type restrictions on $$X(\mu )$$ are required. [ABSTRACT FROM AUTHOR]

Abstract: Spectral properties of strictly singular and disjointly strictly singular operators on Banach lattices are studied. We show that even in the case of positive operators, the whole spectral theory of strictly singular operators cannot be extended to disjointly strictly singular operators. However, several spectral properties of disjointly strictly singular operators are given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the structure of closed algebraic ideals in the algebra of operators acting on a Lorentz sequence space. [Copyright &y& Elsevier]

Abstract: We show that is the only Banach space with unconditional basis that satisfies the equation . This partially improves an old result by Kalton and Peck. We prove that the Kalton–Peck maps are strictly singular on a number of sequence spaces, including for , Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton–Peck type. [Copyright &y& Elsevier]

Abstract: We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families and . [Copyright &y& Elsevier]

We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach-Saks operators, Narrow operators and p-summing operators. [ABSTRACT FROM AUTHOR]

The perturbation classes problem for semi-Fredholm operators asks when the equalities $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$ and $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y)}$$ are satisfied, where $${\mathcal{SS}}$$ and $${\mathcal{SC}}$$ denote the strictly singular and the strictly cosingular operators, and PΦ and PΦ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, $${\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)}$$ if and only if $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y),}$$ and $${\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)}$$ if and only if $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$. Moreover we give examples showing that both direct implications fail in general. [ABSTRACT FROM AUTHOR]

We prove that the component PΦ+(X, Y ) of the perturbation class for the upper semi-Fredholm operators between Banach spaces X and Y coincide with the strictly singular operators when every closed infinite dimensional subspace of X contains an infinite dimensional complemented subspace whose complement is isomorphic to X. Similarly, we prove that the component PΦ•(X, Y ) of the perturbation class for the lower semi-Fredholm operators coincide with the strictly cosingular operators when every infinite codimensional subspace of Y is contained in an infinite codimensional complemented subspace isomorphic to Y . We also give examples of Banach spaces satisfying the aforementioned conditions. [ABSTRACT FROM AUTHOR]

Abstract: We study the perturbation class for the upper semi-Fredholm operators when is non-empty. We show that it coincides with the strictly singular operators when and Y satisfies certain condition, and when and Y is arbitrary. As a consequence, we derive similar results for when . [Copyright &y& Elsevier]

Abstract: We study subspaces of Lorentz spaces and provide an easy-to-check characterization of strictly singular operators defined on these spaces. As an application we obtain stability under duality for the class of strictly singular operators on spaces, extending a theorem of L. Weis for operators on spaces. [Copyright &y& Elsevier]

Abstract: An operator between Banach spaces is said to be finitely strictly singular if for every there exists n such that every subspace with contains a vector x such that . We show that, for , the formal inclusion operator from to is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if then every k-dimensional subspace of contains a vector x with such that for some . [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]