On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.     

SOME CONSEQUENCES OF THE TARSKI-KANTOROVITCH ORDERING THEOREM IN METRIC FIXED POINT THEORY

Abstract

We show that the Tarski-Kantorovitch Principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.     

Critical types of krasnoselskii fixed point theorems in weak topologies

Abstract

In this note, by means of the technique of measures of weak noncom- pactness, we establish a generalized form of fixed point theorem for the sum of T+S in weak topology setups of a metrizable locally convex space, where S is not weakly compact, I?T allows to be noninvertible, and T is not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings.     

A CHARACTERIZATION OF THE BAND OF KERNEL OPERATORS

ABSTRACT

In this note we present a characterization of the band of kernel operators in the abstract setting of Riesz spaces. Under the assumptions that E is an Archimedean Riesz space and F a Dedekind complete Riesz space separated by its ex= tended order continuo88 dual, we obtain a characterization of the band (E_oo ? F)^dd in terms of (sequentially) star or= der continuous operators.     

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Virtual Eigenvalues of the High Order Schrödinger Operator I

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion.     

Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

Let X be a real reflexive Banach space and be maximal monotone. Let be quasibounded, finitely continuous and generalized pseudomonotone with X′⊂D(B), where X′ is a dense subspace of X such that X′∩D(A)≠∅. Let S⊂X∗. Conditions are given under which and intS⊂intR(A+B). Results of Browder concerning everywhere defined continuous and bounded operators B are improved. Extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized (S₊)-condition. Applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains.     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra A_n. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a *-extendible representation σ. A *-extendible representation of A_n is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given *-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of σ(A_n) is weak-* dense in the unit ball of the associated free semigroup algebra if and only if σ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.     

Taylor spectrum of commuting n-contractions

Abstract

In this paper, we characterize the Taylor spectrum for a certain class of commuting n-contractions. We also investigate the behavior of this spectrum under action of involutive automorphisms of the unit ball 𝔹 _n.     

The modulus of matrix semigroups

We treat a result concerning the generator of the modulus semigroup of a strongly continuous semigroup acting on the product of Banach lattices with order continuous norm. Received: 28 March 2003     

A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

IDENTIFICATION OF EXTRAPOLATION SPACES FOR UNBOUNDED OPERATORS

Abstract

Abstract extrapolation spaces for strongly continuous semigroups of linear operators on Banach spaces have been constructed by various methods (see, e.g., [Am (1988)], [DaP-Gr (1984)], [Na (1983)], [Ne (1992)], [Wa (1986)]). Usually they appear as “artefacts” used in some intermediate step in order to solve the Cauchy problem on the original space. Only in a few cases (see the papers by the Dutch school on X ^⊙*, e.g., [Ne (1992)]), and in sharp contrast to the situation for interpolation spaces (see, e.g., [Gr (1969)], [DiB (1991)], [Lu (1985)], [Ac-Te (1987)]), the extrapolation spaces have been identified in a concrete way. It is our intention to fill this gap and subsequently to give an application of the extrapolation method to a perturbation problem.     

Additive local spectrum compressors

Let L(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from L(X) onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from L(X) onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized.     

Characterization of Gaussian semigroups on separable Banach spaces

Let E be a separable real Banach space and denote by BUC (E) the space of bounded and uniformly continuous functions on E. For a C ₀-semigroup acting on BUC(E), we obtain necessary and sufficient conditions ensuring that is Gaussian.     

ISOLATED POINTS OF THE APPROXIMATE POINT SPECTRUM OF CERTAIN LATTICE HOMOMORPHISMS ON Co (X)

Abstract

Suppose X is a locally compact Hausdorff space and C (X) the apace of all continuous complex valued functions on X which vanish at infinity. Let T be a (complex) linear lattice homomorphism on C_o (X) whose adjoint is also a lattice homomorphism. It is sham that every non-zero isolated point of the approximate point spectrum of T lies in the point spectrum of T. An example is given to show that the exclusion of zero is necessary, even when X is compact. The same techniques are then used to show that if also the spectrum of T is finite then T can be written, in a natural manner, as a direct sum of two such lattice homomorphisms; one being an n'th root of an invertible multiplication operator and the other quasi-nilpotent.     

THE o-SPECTRUM OF r-ASYMPTOTICALLY QUASI-FINITE-RANK OPERATORS

Abstract

We introduce the class of r-asymptotically quasi finite rank operators (which properly contains the class of r-compact operators) and show that for these operators the spectrum and o-spectrum coincide.     

Jensen’s inequality for operators without operator convexity

We give Jensen’s inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued continuous convex functions with conditions on the bounds of the operators. We also study operator quasi-arithmetic means under the same conditions.