Tensor Norms and Operator Ideals

The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

This volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area. The only prerequisites are a basic knowledge of functional analysis and measure theory. Features of particular interest include: - A full treatment of the Grothendieck theory of tensor norms; - Coverage of the Chevet-Saphar norms and their duals, along with the associated classes of nuclear, integral and summing operators; - Chapters on the approximation property and the Radon-Nikodym property; - Topics such as the Bochner and Pettis integrals, the principle of local reflexivity and the Grothendieck inequality placed in a natural setting; - The classes of operators generated by a tensor norm and connections with the theory of operator ideals. Each chapter is accompanied by worked examples and a set of exercises, and two appendices provide essential material on summability in Banach spaces and properties of spaces of measures that may be new to the beginner.

In section 1 of this article, a sketch of the life and the career of Klaus Floret is given. Then,in sections 2 to 5, the author reports on the part of the mathematical work of Klaus Floret which was devoted to four important topics in functional analysis in which Klaus worked and in which considerable progress was achieved during his lifetime. These topics are: locally convex inductive limits; bases and approximation properties; - -spaces and the probleme des topologies; tensor norms, operator ideals, spaces of polynomials. Some of the history of these topics is included, sometimes together with an account of some of the leading figures in the field. A prelude, an interlude and a coda present related remarks which do not fit in the main body of the article.

We study the space of $p$-compact operators, $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet–Saphar tensor norm $d_p$ (which is e

In this paper motivated by some recent results of Lewicki and Trombetta, we are interested in studying geometric conditions involving only a set of inequalities, that imply that a closed subspace is one complemented, i.e., it is the range of a projection of norm one. We consider a variation on the notion of best coapproximation, where coapproximation is required for a naturally occurring, smaller set of vectors. We introduce the notion of a weak existence set and show that in the bidual of a Lindenstrauss space, any weak \(^*\)-closed subspace of finite codimension, that is a weak existence set, is one complemented. For an infinite dimensional Banach space \(X\), we consider a not necessarily linear condition, involving the space of compact operators being a semi-\(M\)-ideal in the vector space spanned by the identity operator and the compact operators. Here we show that the identity operator has no best coapproximation in the space of compact operators.

The problem of the special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert-Schmidt kernel is considered. Here is the identity operator and is an integral operator: It is proved that is an eigenvalue of of multiplicity if and only if , where the , , , are bounded operators in of a special structure that are invertible from the left and the right, respectively. Bibliography: 7 titles.

LetEEandFFbe two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent? a) There exists a projection from the spaceL(E,F)L(E,F)of continuous linear operators onto the spaceK(E,F)K(E,F)of compact linear operators. b)L(E,F)=K(E,F)L(E,F)=K(E,F). The answer is positive in certain cases, in particular ifEEorFFhas an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose thatEEandFFare reflexive and thatEEorFFhas the approximation property. Then, ifL(E,F)≠K(E,F)L(E,F)\ne K(E,F), there is no projection of norm 1, fromL(E,F)L(E,F)ontoK(E,F)K(E,F). In this paper, one obtains, in particular, the following result:Theorem.LetFFbe a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such thatF∗F^*has the approximation property. Letλ\lambdabe a real scalar with1>λ>21>\lambda >2. ThenFFcan be equivalently renormed such that, for any projectionPPfromL(F)L(F)ontoK(F)K(F), one has‖P‖≥λ\|P\|\ge \lambda. One gives also various results with two spacesEEandFF.

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Minimal projections onto hyperplanes in vector-valued sequence spaces

In this paper, we focus our study on the fractional homogeneous abstract Cauchy problems of conformable type, wβ(t) = Uw(t), t ∈ [0, T), T ≤ ∞, w(0) = x, where U is a linear operator that generates a fractional β‐semigroup on a Hilbert space H. In fact, we gave an answer to the question if U is the infinitesimal generator of a conformable β‐semigroup {φ(t)}t≥0 of bounded linear operators on a Hilbert space H, under what conditions {φ(t)}t≥0 can be in the Schatten class Cp(H). Moreover, we prove the validity of inequality of resolvent operator R(λ, U) of the operator U in the Schatten norm.

Two-Lipschitz operator ideals

Some ideals of operators on Hilbert space

On Ideals of Operators in Hilbert Space

Let E be a separable Hilbert space and let Γ:Z→(E) be a positive definite map on the additive group Z of the integers whose values are bounded operators on E. Suppose Γ(0)=I, the identity operator on E. According to Naimark dilation theorem there exist a Hilbert space H containing E as a subspace and a unitary operator W on H such that $$\Gamma (n)a=p_{E}W^{n}a,:\ a\varepsilon E, n\varepsilon Z$$ ((1.1)) .KeywordsHilbert SpaceUnit CircleUnitary OperatorInvariant SubspaceMinimality ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Simplicity of linear ordinary differential operators