Compact bilinear operators and paraproducts revisited

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.

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bounded linear operator mapping a reflexive space into X is strictly singular.Remarks, (i) R. S. Phillips exhibited (unpublished) a strictly singular operator which is not compact.Theorem 1 shows that S can indeed be a "much larger" space than the space of compact linear operators.

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.

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

For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.

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 & Co. KGaA, Weinheim)

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.

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.

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.

Let 𝜏 be a faithful normal semifinite trace on a von Neumann algebra ℳ. The block projection operator 𝒫̃︀𝑛 (𝑛 ≥ 2) on the *-algebra 𝑆(ℳ, 𝜏 ) of all 𝜏 -measurable operators is investigated. It is shown that 𝑓(𝒫̃︀𝑛(𝐴)) ≥ 𝒫̃︀𝑛(𝑓(𝐴)) for any operator monotone function 𝑓 on R + and 𝐴 ∈ 𝑆(ℳ, 𝜏 ) + . For an operator convex function 𝑓 on R + , we have 𝑓(𝒫̃︀𝑛(𝐴)) ≤ 𝒫̃︀𝑛(𝑓(𝐴)) for 𝐴 ∈ 𝑆(ℳ, 𝜏 ) + . Conditions are established under which 𝒫̃︀𝑛(𝐴) belongs to the class 𝑆0(ℳ, 𝜏 ) of 𝜏 -compact operators, to the class 𝐹(ℳ, 𝜏 ) of elementary operators, to the classes 𝐿𝑝(ℳ, 𝜏 ) of operators 𝜏 -integrable with 𝑝-th power, or to the ℳ algebra itself. If 𝐴, 𝐵 ∈ 𝑆(ℳ, 𝜏 ) and 𝒫̃︀𝑛(𝐵) is a left (right) inverse for the operator 𝐴, then 𝒫̃︀𝑛(𝐵) is also a left (respectively, right) inverse for the operator 𝒫̃︀𝑛(𝐴).

This work aims to study the Bishop-Phelps-Bollobás Theorem for operators between Banach spaces. We will show that the set of norm-attaining operators from $X$ to $Y$ is norm dense in $\mathcal{L}(X,Y)$ when $ Y$ has property $\beta$ for any Banach space $X$. We will see that the pair $(\ell_1,Y)$ has the Bishop-Phelps-Bollobás property for operators if and only if the Banach space $Y$ has the property $AHSP$. Besides that, if $Y$ is a uniformly convex space, we will see that the pair $(\ell^n_\infty,Y)$ satisfies the Bishop-Phelps-Bollobás property for operators. Furthermore, we will show that if $X$ is a uniformly convex space, the pair $(X,Y)$ has the Bishop-Phelps-Bollobás property for operators for any Banach space $Y$.

Two-Lipschitz operator ideals

Compactness type properties for operators acting in Banach function spaces are not always preserved when the operator is extended to a bigger space. Moreover, it is known that there exists a maximal (weakly) compact linear extension of a (weakly) compact operator if and only if its maximal continuous linear extension to its optimal domain is (weakly) compact. We show that the same happens if we consider AM-compactness for the operator, and we give some partial results regarding Dunford–Pettis operators. Narrow operators—considered as a family defined by a weak compactness type property—are also analyzed from this point of view. Finally, we provide some applications of the fact that an operator from a Banach function space extends to a narrow operator if and only if it is narrow.

Let X X be a (real) Banach space, let D D be an open subset of X X , and let B ( X ) \mathcal {B}(X) denote the collection of all nonempty bounded and closed subsets of X X . Suppose T T is continuous from D ¯ \overline {D} into B ( X ) \mathcal {B}(X) with respect to the Hausdorff metric and strongly pseudo-contractive, while g g is compact from D ¯ \overline {D} into X X . Then T + g T+g has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of D D .

A Banach space ODE framework for inverse scattering via invariant imbedding