Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

Geometric eigenspace of the generator ofC 0-semigroup of positive operators

LetE be a Banach lattice with order continuous norm and {T(t)} _t0 be an eventually compactc ₀-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices.     

Aspects of local spectral theory for positive operators

In this paper we will discuss the local spectral behaviour of a closed, densely defined, linear operator on a Banach space. In particular, we are interested in closed, positive, linear operators, defined on an order dense ideal of a Banach lattice. Moreover, for positive, bounded, linear operators we will treat interpolation properties by means of duality.Dedicated to G. Maltese on the occasion of his 60^th birthday     

Some aspects of the spectral theory of positive operators

We consider various aspects of the following problem: Let T be a positive operator on a Banach lattice such that σ(T)={1}. Does it follow that T≥1?     

Triviality of the peripheral point spectrum

If T_t = e^Zt is a positive one-parameter contraction semigroup acting on l^p(X) where X is a countable set and 1 ≤ p < ∞, then the peripheral point spectrum P of Z cannot contain any non-zero elements. The same holds for Feller semigroups acting on L^p(X) if X is locally compact.     

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).     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by

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

     

Commutators and accretive operators

In this paper, singular values of commutators of Hilbert space operators are estimated. To this aim the accretivity of a transform of the operators is applied. Some recent results of Kittaneh [F. Kittaneh, Singular value inequalities for commutators of Hilbert space operators, Linear Algebra Appl. 430 (2009) 2362-2367] are extended.     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x_n) in X, there exists a subsequence (x_k(n)) so that (Tx_k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if is collectively weakly compact, then M^* is weakly equicompact iff M^** x^**={T^** x^** : T ∈ M} is relatively compact in Y for every x^** ∈X^**; 2) weakly equicompact sets are precompact in for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005     

On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices

Generalizing a recent result of E.B. Davies [4], we show that generators of bounded positive C₀-semigroups on atomic Banach lattices with order continuous norm have trivial peripheral point spectrum. Moreover, we give examples that the peripheral spectrum can be any closed cyclic subset of . Received: 20 September 2005; revised: 23 January 2006     

Kernel estimates for Schrödinger operators

We prove short time estimates for the heat kernel of Schr?dinger operators with unbounded potential in R^N.     

Weakly compact operators and interpolation

The class of weakly compact operators is, as well as the class of compact operators, a fundamental operator ideal. They were investigated strongly in the last twenty years. In this survey, we have collected and ordered some of this (partly very new) knowledge. We have also included some comments, remarks and examples.     

The closed ideal of (c o ,p,q)-summing operators

For 1/p+1/q1, we study the closed ideal formed by the (c _o ,p,q)-summing operators. It turns out thatT:XY does not belong to if and only if it factors the mapId:l _p ^*l _q . By localization, we get the ideal that consists of those operatorsT for which all ultrapowersT ^u are contained in . Operators in the complement of are characterized by the property that they factor the mapsId:l _p ^*n l _q ⁿ uniformly. Our main tools are ideal norms.Supported by DFG grant PI 322/1-2     

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.     

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH ₂, and the Laplacian operator onL ₂((), for any bounded open subset of ². Furthermore, we show that these operators are chaotic, in the sense of Devaney.