Composition operators on Banach function spaces

We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a -finite measure space, Lorentz function spaces on a -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.

     

Power bounded operators and supercyclic vectors

By the well-known result of Brown, Chevreau and Pearcy, every Hilbert space contraction with spectrum containing the unit circle has a nontrivial closed invariant subspace. Equivalently, there is a nonzero vector which is not cyclic.

We show that each power bounded operator on a Hilbert space with spectral radius equal to one has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant positive cone.

     

On the ideal-triangularizability of positive operators on Banach lattices

There are some known results that guarantee the existence of a nontrivial closed invariant ideal for a quasinilpotent positive operator on an -space with unit or a Banach lattice whose positive cone contains an extreme ray. Some recent results also guarantee the existence of such ideals for certain positive operators, e.g. a compact quasinilpotent positive operator, on an arbitrary Banach lattice. The main object of this article is to use these results in constructing a maximal closed ideal chain, each of whose members is invariant under a certain collection of operators that are related to compact positive operators, or to quasinilpotent positive operators.

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Some descriptive set-theoretic properties of the isomorphism relation between Banach spaces

Consider the space of closed linear subspaces of a separable Banach space equipped with the standard Effros Borel structure. The isomorphism relation between Banach spaces being elements of  determines a partition of  . In this note we prove a result describing the complexity of analytic subsets of  intersecting a large enough number of the above-mentioned parts of  .

     

On quasispectral maximal subspaces of a class of Volterra-type operators

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladi\v{c} in 1984. As an application a class of quasinilpotent operators on -spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.

     

Interpolating hereditarily indecomposable Banach spaces

The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator admits a factorization through a H.I. space. The same result holds for every strictly singular operator .

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.     

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).     

On positive unipotent operators on Banach lattices

Let be a positive operator on a complex Banach lattice. We prove that is greater than or equal to the identity operator if

     

Characterizations of Disjointness preserving operators on vector-valued function spaces

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator is compact (resp. completely continuous) if and only if

for all

where is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and is compact (resp.  is uniformly completely continuous).

     

Invariant subspaces for polynomially hyponormal operators

We show that if is a bounded operator on a Hilbert space such that for every polynomial , then has a nontrivial invariant subspace.

     

Compact sets of compact operators in absence of

We characterize the compactness of a subset of compact operators between Banach spaces when the domain space does not have a copy of

     

Hyperinvariant subspaces for some subnormal operators

In this article we employ a technique originated by Enflo in 1998 and later modified by the authors to study the hyperinvariant subspace problem for subnormal operators. We show that every ``normalized'subnormal operator such that either does not converge in the SOT to the identity operator or does not converge in the SOT to zero has a nontrivial hyperinvariant subspace.

     

Weyl's theorem for perturbations of paranormal operators

A bounded linear operator on a Banach space is said to satisfy ``Weyl's theorem' if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if is a paranormal operator on a Hilbert space, then satisfies Weyl's theorem for every algebraic operator which commutes with .

     

Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic Banach space. The construction of this operator relies on the existence of transfinite -spreading models in the dual of the space.

     

Spectrum of interpolated operators

Let be a compatible pair of Banach spaces and let be an operator that acts boundedly on both and . Let be the corresponding operator on the complex interpolation space .

The aim of this paper is to study the spectral properties of . We show that in general the set-valued function is discontinuous even in inner points and show that each operator satisfies the local uniqueness-of-resolvent condition of Ransford. Further we study connections with the real interpolation method.

     

Products of EP operators on Hilbert spaces

A Hilbert space operator is called the EP operator if the range of is equal to the range of its adjoint . In this article necessary and sufficient conditions are given for a product of two EP operators with closed ranges to be an EP operator with a closed range. Thus, a generalization of the well-known result of Hartwig and Katz (Linear Algebra Appl. 252 (1997), 339-345) is given.

     

On the Helton class of -hyponormal operators

In this paper we show that the Helton class of -hyponormal operators has scalar extensions. As a corollary we get that each operator in the Helton class of -hyponormal operators has a nontrivial invariant subspace if its spectrum has its interior in the plane.

     

On Beurling-type theorems in weighted and Bergman spaces

We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted and Bergman spaces.

     

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five -th roots of the identity for every 2$">. For invertible normal operators four factors suffice in general.