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.     

A Dodds–Fremlin Property for Asplund and Radon–Nikodým Operators

Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator.     

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.      

Sums of Order Bounded Disjointness Preserving Linear Operators

Siberian Mathematical Journal - Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is...     

Analysis of Non-normal Operators via Aluthge Transformation

Let T be a bounded linear operator on a complex Hilbert space . In this paper, we show that T has Bishops property () if and only if its Aluthge transformation has property (). As applications, we can obtain the following results. Every w-hyponormal operator has property (). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chs problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re(T) = (Re T) or not.     

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

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     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L_p(0,1) and C_p, are non-V-spaces if p 2; and that the spaces l_p are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems.     

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T_a⁽⁾ acting on weighted Bergman spaces over the unit disk , in dependence on , and compare their limit behavior under with corresponding properties of the initial symbol a.     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

On Order Bounded Subsets of Locally Solid Riesz Spaces

In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.     

Spectra of Composition Operators on BMOA

It is shown that if ϕ is a univalent self-map on the unit disk is not an automorphism and has a fixed point in and if the essential spectral radius of the composition operator C_ϕ on H² is different from zero, then the spectrum of C_ϕ on BMOA coincides with This answers in the affirmative a conjecture by MacCluer and Saxe.     

Solutions of Two Problems in the Theory of Disjointness Preserving Operators

In this note, our aim is to solve two problems in the theory of disjointnesss preserving operators which are posed by Abramovich and Kitover using some examples constructed by the same authors.     

Measures with Bounded Variation with Respect to a Normed Ideal of Operators and Applications

We introduce in a natural way the notion of measure with bounded variation with respect to a normed ideal of operators and prove that for each maximal normed ideal of operators (, ), is true the following result: If U ∈ L(C(T,X), Y) with G the representing measure of U and G : Σ → ((X, Y),) has bounded variation, then U ∈ (C(T,X), Y). As an application of this result we prove that an injective tensor product of an integral operator with an operator belonging to a maximal normed ideal of operators (,) belongs also to (, ).     

Spectra of Compact Composition Operators Over Bounded Symmetric Domains

Let H²(D) denote the Hardy space of a bounded symmetric domain in its standard Harish-Chandra realization, and let be the weighted Bergman space with and where is a critical value depending on D. Suppose that is holomorphic. We show that if the composition operator defined by is compact (or, more generally, power-compact) on H²(D) or then has a unique fixed point z₀ in D. We then prove that the spectrum of as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluers previous spectrum results on the unit ball B_n extend to H^p(n) (not only for p = 2 but for all p > 1) and (for p 1), where ⁿ is the polydisk in      

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.