Spectrum for compact operators on Banach spaces

It is an old problem in operator theory whether a pair of norm one compact Hermitian operators with "small" (in norm) commutator can be "well" approximated by a commuting pair of Hermitian operators.We show that, for operators of rank not exceeding n, such approximants exist provided ||[A, B]||/n 1/2 is small.This improves a result of Pearcy and Shields and sheds some new light on the original question and its relationship to a few related ones.

introduces the following concept of a strictly singular operator in [3].Definition.Let X and Y be Banach spaces and let T be a bounded linear operator mapping X into F. T is said to be strictly singular if given any infinite dimensional subspace M of X, T restricted to M is not an isomorphism (i.e., linear homeomorphism).In particular, every compact operator is strictly singular.Kato proceeds to show that the space S of strictly singular operators possesses some of the important features of the space of compact operators, e.g., S is a closed subspace of the space of bounded linear operators from A to F. If X= Y, then S is a closed ideal.If X and F are Hubert spaces, then every strictly singular operator from A to F is compact.The following two questions, posed by Kato, are answered below.(1) Is every strictly singular operator compact?(2) Is the conjugate of a strictly singular operator strictly singular?Theorem.3 (a) Every bounded linear operator from k to lp or from lp to l2, l^p9i2, oo is strictly singular.(b) Let X be a Banach space which does not contain an infinite dimensional reflexive subspace.Then every bounded linear operator mapping X into a reflexive space is strictly singular.Also any bounded linear operator mapping a reflexive space into X is strictly singular.Remarks, (i) R. S. Phillips exhibited (unpublished) a strictly singular operator which is not compact.Theorem 1 shows that S can indeed be a "much larger" space than the space of compact linear operators.

The injective and surjective hulls of the ideal of (p,q)-compact operators and their approximation properties

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this article, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a) for non-autonomous equations with finite delays and uniformly bounded compact coefficient operators in Banach spaces and (b) for Volterra difference equations with infinite delay in finite dimensional spaces.

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples. We then prove compactness of bilinear paraproducts with CMO symbols by combining one of the properties of compact bilinear operators thus obtained with vanishing Carleson measure estimates and interpolation of bilinear compactness.

This research paper offers a comprehensive investigation into the concept of norm-attainability in Banach and Hilbert spaces. It establishes that norm-attainable operators exist if and only if the target space is a Banach space and that norm-attainable polynomials are inherently linear. In convex optimization scenarios, norm-attainable polynomials lead to unique global optima. The paper explores the norm of norm-attainable operators, revealing its connection to supremum norms. In Hilbert spaces, norm-attainable operators are self-adjoint. Additionally, it shows that in finite-dimensional spaces, all bounded linear operators are norm-attainable. The research also examines extremal polynomials and their relationship with derivative roots, characterizes optimal solutions in norm-attainable operator contexts, and explores equivalence between norm-attainable operators through invertible operators. In inner product spaces, norm-attainable polynomials are identified as constant. Lastly, it highlights the association between norm-attainable operators and convex optimization problems, where solutions lie on the unit ball's boundary. This paper offers a unified perspective with significant implications for functional analysis, operator theory, and optimization in various mathematical and scientific domains.

It is shown that there is a nonseparable Banach space on which every operator has the form S + pi where p is a scalar and S is an operator with separable range.An open problem in the theory of Banach spaces is the construction of a Banach space with the property that all operators from it to itself are the sum of a multiple of the identity and a compact operator.Since compact operators have separable range a related question is to find a (nonseparable) Banach space on which all operators are the sum of a multiple of the identity and an operator with separable range.This was first done by Shelah using an extra set-theoretic axiom in [1].It is the purpose of this paper, which is the continuation of research which began in [3], to prove the same result without appealing to axioms other than those of ZFC.The following notation will be used.The integers will be considered as ordinals so that n Gm means the same as n < m.The set of integers is denoted as w instead of N and the first uncountable ordinal is wi.The symbols [X]x and [X]<A denote, respectively, the set of A sized subsets of X and less than A sized subsets of X.The set of functions from Y to X is denoted YX.In particular, WlR, when endowed with the usual pointwise operations, is the real vector space of "wi-tuples".For X ui the characteristic function of X will be denoted by lx-Let us begin by reviewing the construction from [2] of a Banach space on which every operator is the sum of an operator with separable range and one which is diagonal with respect to a fixed basis.Let 21 [u>i]<N be such that:i. ua = wi.2. 21 is closed under subsets.It is possible to define a function || ||a on ">R by ||/||*=sup{{|/(a)|;aGA};AG2l}.Let B = {UlR; ||/||a is finite} and note that (B, || ||a) is a normed linear space.Of course if 21 = [wi]<"' then B = /i(wi) but we will assume that there is a family 53 [wi]<w such that: 3. 93 is closed under subsets.4. an8c [uji]1.5.For all families of disjoint pairs {{a(a),b(a)}; a G u>i} and integers k there is X G [ui]k such that {a(a); a G X} G 21 and {6(a); a G X} G 03.

We consider linear operator equations of the form where $K:{\cal X}\to{\cal Y}$ is a compact linear operator between Hilbert spaces ${\cal X} \hbox{ and } {\cal Y}.$ We assume y to be attainable, i.e., that problem (1) has a solution x† = K†y of minimal norm. Here K† denotes the (Moore‐Penrose) generalized inverse operator of K, which is unbounded when K is compact, with infinite dimensional range. Hence problem (1) is ill‐posed and has to be regularized in order to compute a numerical solution. We want to approximate the solution x† of the equation (1), when only an approximation yδ of y is available with ‖yδ − y‖ ≤ δ, where δ is called the noise level. Since K†yδ is not a good approximation of x†, we approximate x† with xδα := Rαyδ where {Rα} is a family of continuous operators depending on a parameter α. A classical example is the Tikhonov regularization defined by Rα = (K*K + αI)−1 K*, where I denotes the identity and K* the adjoint of K.Recently, new Tikhonov based regularization methods have been proposed in [1], [2] and [3], under the name of fractional Tikhonov, to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order.In this paper, we firstly provide a saturation result similar to the well‐known saturation result for Tikhonov regularization [6]: indeed, Tikhonov regularization under suitable a‐priori assumption and a‐priori choice rule, α = α(δ) ∼ c(δ)2/3, is of optimal order and the best possible convergence rate obtainable is ‖xδα−x†‖ = O(δ2/3). On the other hand, let R(K) be the range of K and let Q be the orthogonal projector onto $\overline{R(K)},$, if then x† = 0, as long as R(K) is not closed, and this shows how Tikhonov regularization for an ill‐posed problem with compact operator never yields a convergence rate which is faster than O(δ2/3), since it saturates at this rate. Such results motivated us to introduce the iterated versions of fractional Tikhonov methods in the same spirit of the iterated Tikhonov method. We prove that those iterated methods can overcome the afore‐mentioned saturation results.Afterwards, inspired by the works [4,5] we introduce the nonstationary variants of our iterated methods. Differently from the nonstationary iterated Tikhonov, we have two nonstationary sequences of parameters. In the noise free case, we give sufficient conditions on these sequences to guarantee the convergence providing also the corresponding convergence rates. In the noise case, we show the stability of the proposed iterative schemes proving that they are regularization methods. For the brief space we have at disposal, we will only cite some of the main results we obtained for one kind of fractional Tikhonov algorithms, that we call weighted Tikhonov. For the results about the other fractional Tikhonov algorithm analysis, proofs, numerical examples and full details, we invite the interested reader to look at the full paper [7]. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Numerical peak holomorphic functions on Banach spaces

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈ ((lp)T, l∞), where T is an arbitrary triangle, 1 ≤ p ≤ ∞, to be a compact operator. In the past, only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces.

Our aim is to solve the problem asked by Bahramnezhad and Azar in "KB-operators on Banach lattices and their relationships with Dunford-Pettis and order weakly compact operators". We show that a continuous operators R from a Banach lattice N into a Banach space M is a b-weakly compact operator if and only if R is a KB-operator.

In this chapter we address the following subjects: the spectrum and eigenvalues of an operator; the resolvent and non-emptyness of the spectrum; finite-rank operators, the approximation property and compactness in Banach spaces; compactness criteria for sets in specific spaces; definition and properties of compact operators; operators of the form \(I - T\) with T a compact operator; the structure of the spectrum of a compact operator.