The metric approximation property,norm-one projections and intersection properties of balls

We show that ifE is a Banach space with the Radon-Nikodym property thenE has the metric approximation property if and only if the space of finite rank operators is locally complemented in the space of bounded operators.     

New criterion of the approximation property

This paper is concerned with the approximation property which is an important property in Banach space theory. We show that a Banach space X has the approximation property if (and only if), for every Banach space Y, the set of finite rank operators from X to Y is dense in the corresponding space of compact operators, in the usual topology of uniform convergence on compact sets.     

一致可逆性质及广义(ω)性质的摄动

Hilbert空间算子T∈B(H)称为是一致可逆的,若对任意的S∈B(H),TS与ST的可逆性相同.本文中根据一致可逆性质定义了一个新的谱集,用该谱集来研究广义(ω)性质的稳定性,即考虑了Hilbert空间上有界线性算子的有限秩摄动、幂零摄动以及Riesz摄动的广义(ω)性质.之后研究了能分解成有限个正规算子乘积的一类算子的广义(ω)性质的稳定性.     

On Bounded Local Resolvents

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum.     

Property (R) for Bounded Linear Operators

We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid, or left polaroid, operators.     

Operators which have a closed quasi-nilpotent part

We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property.

     

The Approximation Property and Locally Convex Spaces Defined by the Ideal of Approximable Operators

The connection between the approximation property and certain classes of locally convex spaces associated with the ideal ?? of approximable operators will be discussed. It will be shown that a Frechet Montel space has the approximation property iff it is a mixed ??-space, or equivalently, iff its strong dual is a ??-space. Similarly, a Silva space has the approximation property iff it is a mixed ??-space or equivalently iff it is a ??-space. We shall also show that a Frechet Schwartz space with the bounded approximation property is always a ??-space.     

The repetition property for sequences on tori generated by polynomials or skew-shifts

The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schrödinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schrödinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.     

Decomposition of the Space of Regular Operators

Let E be a directed vector space with the Riesz decomposition property and F be a complete v?ctor lattice. It is proved that if G is a solid subspace of E then the subspace of all regular operators annihilating on G is a component in the space of all regular operators from E into F.¹²     

Carathéodory-Fejér interpolation in locally convex topological vector spaces

We study Carathéodory-Herglotz functions whose values are continuous operators from a locally convex topological vector space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself.     

The single-valued extension property for sums and products of commuting operators

It is shown that the sum and the product of two commuting Banach space operators with Dunford's property (C) have the single-valued extension property.     

Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970s. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.     

Kruglov operator and operators defined by random permutations

The Kruglov property and the Kruglov operator play an important role in the study of geometric properties of r. i. function spaces. We prove that the boundedness of the Kruglov operator in an r. i. space is equivalent to the uniform boundedness on this space of a sequence of operators defined by random permutations. It is also shown that there is no minimal r. i. space with the Kruglov property.     

Very smooth points of spaces of operators

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is anM-ideal in the space of bounded operators, a very smooth operatorT attains its norm at a unique vectorx (up to a constant multiple) andT(x) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon-Nikodym property. We give an example of a smooth Banach space without any very smooth points.     

Stability of abstract nonlinear nonautonomous differential-delay equations with unbounded history-responsive operators

We consider a class of nonautonomous functional-differential equations in a Banach space with unbounded nonlinear history-responsive operators, which have the local Lipshitz property. Conditions for the boundedness of solutions, Lyapunov stability, absolute stability and input-output one are established. Our approach is based on a combined usage of properties of sectorial operators and spectral properties of commuting operators.     

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.     

Symbolic Calculus and Fredholm Property for Localization Operators

We study the composition of time-frequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operator possesses the Fredholm property, and thus its range is closed in the target space.     

Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators

First we show that every real Banach space satisfying a certain property, calledβ (used by Lindenstrauss and Partington) verifies the denseness of the numerical radius attaining operators. Using this result and a renorming theorem by Partington we conclude that every Banach space is isomorphic to a new one satisfying the denseness of the numerical radius attaining operators.     

Numerical Radius Attaining Operators and the Radon-Nikodym Property

We prove that, for any Banach space X, the set of operatorson X whose adjoints attain their numerical radii is dense inthe space of all operators. We also show the denseness of theset of numerical radius attaining operators on a Banach spacewith the Radon-Nikodym property.     

Powers and direct sums

This paper investigates when a bounded operator on Hilbert space has the property that its square is similar to to the direct sum of it with itself. A few general results are obtained and the normal operators and unilateral shifts having this property are characterized. Several additional examples are explored.