Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Continuity of the Restriction of C 0-Semigroups to Invariant Banach Subspaces

A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for strong continuity of the restriction of a C₀-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators are bounded near the initial time, then the restricted semigroup is strongly continuous.     

Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions

We consider Hilbert spaces of analytic functions defined on an open subset of , stable under the operator M_u of multiplication by some function u. Given a subspace of which is nearly invariant under division by u, we provide a factorization linking each element of to elements of on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to and u(z) = z, we obtain interesting results involving a H²-norm control. In particular, we deduce a factorization for the kernel of Toeplitz operators on Dirichlet spaces. Finally, we give a localization for the problem of extraneous zeros.Submitted: January 18, 2003 Revised: December 20, 2003     

Polar Decompositions of C 0(N) Contractions

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C₀(N) contraction if and only if , where U is a singular unitary operator with multiplicity and x₁, . . . , x_d are orthonormal vectors satisfying . For a C₀(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.     

Positive one-parameter semigroups on ordered banach spaces

In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.First we discuss the general structure of ordered Banach spaces and their ordered duals. We examine normality and generation properties of the cones of positive elements with particular emphasis on monotone properties of the norm. The special cases of Banach lattices, order-unit spaces, and base-norm spaces, are also examined.Second we develop the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak^*-continuous semigroups on the dual spaces. Initially we derive analogues of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently we analyse strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.     

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space and a suitable closed subspace of , the Schur complement of A to is defined. The basic properties of are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. To the memory of Professor Mischa Cotlar     

Decomposition Methods in Banach Spaces via Supplemented Subspaces Resembling Pełczyński’s Decomposition Methods

Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X ^m is isomorphic to Yⁿ for some m, n ∈ IN^*. So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pełczyński’s decomposition methods. In order to do this, we introduce several notions of Schroeder–Bernstein Quadruples acting on the spaces X, Y, A and B. Thus, we characterize them by using some Banach spaces recently constructed. Received: October 4, 2005.     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Common Hypercyclic Subspaces

We give a criterion for a family of operators to have a common hypercyclic subspace. We apply this criterion to a family of homotheties.     

Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

We investigate properties of subspaces of L₂ spanned by subsets of a finite orthonormal system bounded in the L_∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L₁ and the L₂ norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L₂ⁿ, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin’s splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ_p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L₂ and the L_p norms are close (again, up to a logarithmic factor). Partially supported by an Australian Research Council Discovery grant. This author holds the Canada Research Chair in Geometric Analysis.     

Positive operators on Krein spaces

A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literature. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace. Dedicated to the memory of M.G. Krein (1907–1989)     

Extension of Locally Defined Indefinite Functions on Ordered Groups

We give a definition of -indefinite function of archimedean type, on an interval of an ordered group with an archimedean point. We say that has the indefinite extension property if every continuous -indefinite function of archimedean type, on an interval of , can be extended to a continuous -indefinite function on the whole group .We show that if a group is semi-archimedean and it has the indefinite extension property, then with the lexicographic order and the product topology has the indefinite extension property. As a corollary it is obtained that the groups and with the lexicographic order and the usual topologies, have the indefinite extension property.To Professor José R. León, for his kind encouragement.Both authors were supported in part by the CDCH of the Univ. Central de Venezuela and by CONICIT grant G-97000668. Both authors were visitors at IVIC during the realization of this paper.Submitted: August 8, 2002 Revised: January 30, 2003     

Norms of Toeplitz and Hankel Operators on Hardy Type Subspaces of Rearrangement-Invariant Spaces

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.     

A characterization of boundary conditions yielding maximal monotone operators

We provide a characterization for maximal monotone realizations for a certain class of (nonlinear) operators in terms of their corresponding boundary data spaces. The operators under consideration naturally arise in the study of evolutionary problems in mathematical physics. We apply our abstract characterization result to Port–Hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco-elasticity.     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if

then there is a linear isometry U : X → Y so that

where is defined by

Representation properties of coarse isometries in free ultrafilter limits on are also discussed.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.