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.     

Norm attaining operators and renormings of Banach spaces

We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc _o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.     

Convex numerical radius

In this work we introduce the concept of convex numerical radius for a continuous and linear operator in a Banach space, which generalizes that of the classical numerical radius. Besides studying some of its properties, we give a version of James's sup theorem in terms of convex numerical radius attaining operators.     

A counterexample on numberical radius attaining operators

We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Our result was conjectured by C. Cardassi in 1985.     

A characterization of reflexive spaces

We prove that a Banach space is reflexive if for every equivalent norm, the set of norm attaining functionals has non-empty norm-interior in the dual space. It is also proved that the set of norm attaining functionals on a Banach space that is not a Grothendieck space is not a w*-G _δ subset of the dual space.     

Additions to the Periodic Decomposition Theorem

A pair of linear bounded commuting operators T₁, T₂ in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T₁)(I-T₂) = Ker (I-T₁) + Ker (I-T₂). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Révész that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Best Approximation in Numerical Radius

Let X be a reflexive Banach space. In this article, we give a necessary and sufficient condition for an operator T ∈ 𝒦(X) to have the best approximation in numerical radius from the convex subset 𝒰 ? 𝒦(X), where 𝒦(X) denotes the set of all linear, compact operators from X into X. We also present an application to minimal extensions with respect to the numerical radius. In particular, some results on best approximation in norm are generalized to the case of the numerical radius.     

Norm or Numerical Radius Attaining Multilinear Mappings and Polynomials

We study the denseness or norm of numerical radius attainingmultilinear mappings and polynomials between Banach spaces,and examine the relations between norms and numerical radiiof such mappings.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

Norm attaining operators fromL 1 intoL ∞

We show that the set of norm attaining operators is dense in the space of all bounded linear operators fromL ₁ intoL _∞. Partially supported by Human Capital and Mobility. Project No. ERB4050Pl922420, Geometry of Banach spaces. Supported by D.G.I.C.Y.T., Project No. PB93-1142.     

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.     

Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in (B _X; X) is the supremum of the set

SPECTRAL PROPERTIES OF POSITIVE ELEMENTS IN BANACH LATTICE ALGEBRAS

Abstract

We prove that if a unital Banach lattice algebra has sufficiently many one-dimensional elements and if its unit element has sufficiently many components then its positive elements have spectral properties analogous to those of positive operators on Banach lattices. In particular, if a positive element is irreducible (in the sense that (1—e)xe > 0 for all components e of 1 satisfying 0 ≠ e ≠ 1) and compact, its spectral radius is positive and its spectrum shows cyclic behaviour.     

On maximal monotone operators with relatively compact range

It is shown that every maximal monotone operator on a real Banach space with relatively compact range is of type NI. Moreover, if the space has a separable dual space then every maximally monotone operator T can be approximated by a sequence of maximal monotone operators of type NI, which converge to T in a reasonable sense (in the sense of Kuratowski-Painleve convergence).     

A Version of James' Theorem for Numerical Radius

We prove that if all the rank-one bounded operators on a Banachspace X attain their numerical radii, then X must be reflexive,but the converse does not hold. In fact, every reflexive spacewith basis can be renormed in such a way that there is a rank-oneoperator not attaining the numerical radius. 1991 MathematicsSubject Classification 47A12, 46B10.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Invariant subspaces of positive strictly singular operators on Banach lattices

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.     

Banach空间中带扰动的m-增生算子的零点与映象定理

设X为实Banach空间,TX D(T)→2x为m-增生算子,CD(T)→X为有界算子(未必连续),而C(T+I)-1为紧算子.假设     

Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.