Generic existence and uniqueness of positive eigenvalues and eigenvectors

We consider a closed cone of positive operators on an ordered Banach space and prove that a generic element of this cone has a unique positive eigenvalue and a unique (up to a positive multiple) positive eigenvector. Moreover, the normalized iterations of such a generic element converge to its unique eigenvector.     

Approximation of p-multiplier Operators via their Spectral Projections

Conditions for a p-multiplier $\psi: {\mathbb{Z}} \to {\mathbb{C}}Conditions for a p-multiplier are presented which ensure that the corresponding operator T_ψ, acting in , can be approximated by linear combinations of p-multiplier projections coming from the uniform operator closed, unital algebra of operators generated by T_ψ. Functions of bounded variation on play an important role, as do certain Λ (p)-sets. Dedicated to the memory of H. H. Schaefer Werner J. Ricker: Former Alexander von Humboldt Fellow at the Universit?t Tübingen, hosted by Prof. H.H. Schaefer from Sept. 1987 – Feb. 1988.     

Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,H ^s,p , which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S – 1/p . For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.This research was supported by Junta Nacional de Investigação Científica e Tecnológica (Portugal) and the Bundesminister für Forschung und Technologie (Germany) within the projectSingular Operators-new features and applications, and by a PRAXIS XXI project under the titleFactorization of Operators and Applications to Mathematical Physics.     

On a class of stably finite nuclear C*-algebras

In this note we show that a separable C^*-algebra is nuclear and has a quasidiagonal extension by (the ideal of compact operators on an infinite-dimensional separable Hilbert space) if and only if it is anuclear finite algebra (NF-algebra) in the sense of Blackadar and Kirchberg, and deduce that every nuclear C^*-subalgebra of aNF-algebra isNF. We show that strongNF-algebras satisfy a Følner type condition.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

The Canonical Spectral Measure in Köthe Echelon Spaces

Operator and measure theoretic properties of the canonical spectral measure acting in K?the echelon sequence spaces X are characterized via topological and geometric properties of X (such as being nuclear, Montel, satisfying the density condition, etc.).     

The functional model of a j-unitary node with a given j-inner characteristic matrix function

The known model of aj-unitary node with a givenj-inner characteristic matrix functionW is obtained from the known model of a unitary node with a given bi-inner characteristic matrix functionS, using the Potapov-Ginzburg transforms of the nodes and their characteristic functionsW andS. We show that some new properties of the L. de Branges reproducing kernal Hilbert spaces (W) which were discovered by H. Dym are characteristic properties of these spaces.     

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.     

Generic convergence of infinite products of positive linear operators

We present several results concerning the asymptotic behavior of (random) infinite products of generic sequences of positive linear operators on an ordered Banach space. In addition to a weak ergodic theorem we also obtain convergence to an operator of the formf(·) wheref is a continuous linear functional and is a common fixed point.     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

An alternative proof of a Radon-Nikodym theorem for lattice homomorphisms

We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.     

ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS

ABSTRACT

We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.     

Some remarks on disjointness preserving operators

In this note we present a simple proof of the following results: if T: E E is a lattice homomorphism on a Banach lattice E, then: i) (T)={1} implies T=I; and ii) r(T–I)<1 implies TZ(E), the center of E.     

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra     

EQUIMEASURABLE SETS IN RIESZ SPACES

Abstract

We discuss the notion of equimeasurability in the general setting of Riesz spaces and obtain a characterization for (Carleman) abstract kernel operators in terms of equimea=surable sets.     

Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an l_p-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on l_p that is quasinilpotent at a non-zero positive vector x₀ has a non-trivial invariant closed subspace. In this paper, it is proved that if is a collection of continuous operators with moduli on l_p that is finitely modulus-quasinilpotent at a non-zero positive vector x ₀ then and its right modulus sub-commutant have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l _p whose moduli commute with a non-zero positive operator I on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace, so that all positive operators on l _p which commute with a non-zero positive operator S on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace. This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147). Received: 4 December 2005 Revised: 19 June 2006     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

MULTIPLIERS OF MATRIX SPACES

Abstract

Suppose X and Y are FK spaces in which ? the span of the coordinate vectors (e_n) is dense. Let L(X,Y) denote the space of all matrices of the form E_i(T(e_j)) as T ranges over all continuous linear operators from X into Y; here e_i represents the i^th coordinate vector and E_i represents the i^th coordinate functional. Let M(L(X, Y)) denote the space of all matrices B such that (B(i,j)A(i,j)) is in L(X,Y) whenever A is in L(X,Y). In this paper we shall show how the summability properties of X and Y determine the extent of M(L(X,Y)) and conversely how the extent of M(L(X,Y)) determines the summability properties of both X and Y.