The relationship between almost Dunford-Pettis operators and almost limited operators

We investigate Banach lattices on which each positive almost Dunford- Pettis operator is almost limited and conversely.     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

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.      

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.Research supported by the Australian Research Council     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.     

On the Riesz representation theorem and integral operators

We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.     

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)     

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.     

Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions

Sarason interpolation and Toeplitz corona problems are studied for almost periodic matrix functions. Recent results on almost periodic factorization and related generalized Toeplitz operators are the main tools in the study.Supported in part by NSF Grant DMS 9500912Supported in part by NATO Collaborative Research Grant 950332Supported by NSF Grant DMS 9500924     

Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators

LetA andB be two anticommuting self-adjoint operators andV() be a symmetric operator in a Hilbert space, where >0 is a parameter. It is proven that, under some conditions forV(), the resolvents of A+² B±²|B|+V() converge as . Applications to the nonrelativistic-limit problem of Dirac operators and supersymmetry are discussed.This work is supported by the Grant-In-Aid 0560139 for science research from the Ministry of Education, Japan.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Commutation properties of anticommuting self-adjoint operators,spin representation and Dirac operators

LetA andB be anticommuting self-adjoint operators in a Hilbert space . It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB. LetU _s be the partial isometry associated with the self-adjoint operatorsS, i.e., the partial isometry defined by the polar decompositionS=U _S |S|. LetP _S be the orthogonal projection onto (KerS). Then the following are proven: (i) The operatorsU _A ,U _B ,U _C(A,B) ,P _A ,P _B , andP _A P _B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra of the special unitary groupSU(2); (ii) There exists a Lie algebra associated with those operators; (iii) If is separable andA andB are injective, then gives a completely reducible representation of with each irreducible component being the spin representation of the Clifford algebra associated with ³; This result can be extended to the case whereA andB are not necessarily injective. Moreover, some properties ofA+B are discussed. The abstract results are applied to Dirac operators.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

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     

On unbounded hyponormal operators II

The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.     

Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps.These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from in terms of their symbols.     

Bounded tightness for weak topologies

In every Hausdorff locally convex space for which there exists a strictly finer topology than its weak topology but with the same bounded sets (like for instance, all infinite dimensional Banach spaces, the space of distributions or the space of analytic functions in an open set , etc.) there is a set A such that 0 is in the weak closure of A but 0 is not in the weak closure of any bounded subset B of A. A consequence of this is that a Banach space X is finite dimensional if, and only if, the following property [P] holds: for each set and each x in the weak closure of A there is a bounded set such that x belongs to the weak closure of B. More generally, a complete locally convex space X satisfies property [P] if, and only if, either X is finite dimensional or linearly topologically isomorphic to . Received: 11 June 2003     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

On the peripheral spectrum of monic operator polynomials with positive coefficients

I wish to thank R. Nagel for his guidance and suggestions in the preparation of this paper. Also, I would like to thank G. Greiner and F. Räbiger for many interesting and helpful discussions.     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).