Manfred Wolff

On lattice isomorphisms with positive real spectrum and groups of positive operators

One basic problem in the spectral theory of bounded operators is the following: Which additional properties of an operator T imply it to be the identity when the spectrum a(T) of T is known to consist of the element 1 only? For example, if T is a normal operator on Hilbert space then T=I iff o-(T)={1}. More recently, Akeman and Ostrand [1] proved the same result for automorphisms of C*-algebras, and Kamowitz, Scheinberg [4], and Johnson [3] proved it for commutative semisimple Banach algebras. It is one of our aims to prove an analogous result for lattice isomorphisms of a Banach lattice E. In fact, we show the following more general result to hold: A lattice isomorphism T of E has positive real spectrum iff it belongs to the center Z(E) of E. For concrete function lattices (e.g. C(X) or LP(X, #)) this means that T is multiplication by a positive function. This general result enables us to prove a generalization of a result of Lotz [5] on the discreteness of groups of positive operators. In addition, we show that a C0-semi-grou p of lattice homomorphisms on E which possesses a bounded generator, is contained in the center Z(E). In Section 1 we explain some notations and give some auxiliar results. The second section contains our main theorem, the third being reserved for the applications mentioned above.

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 (Sn) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onSn andT the peripheral spectra πσ(Sn) ofSn converge to the peripheral spectrum πσ(T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.

Spectral and asymptotic properties of dominated operators

We investigate the relationship between the peripheral spectrum of a positive operator T on a Banach lattice E and the peripheral spectrum of the operators 5 dominated by T, that is, \Sx\ < T\x\ for all x e E. This can be applied to obtain inheritance results for asymptotic properties of dominated operators.

Spectral theory of group representations and their nonstandard hull

We construct the nonstandard hull of a not necessarily bounded strongly continuous representationU of the locally compact semigroupS on a Banach spaceE. Then we apply our results to the theory of the spectrum σ (U) ofU, mainly in cases whereS is an abelian group, e.g.S=R. First of all we obtain generalizations to the unbounded case of results known for the bounded one. Secondly we introduce the notion of the Riesz part R σ(U) of σ(U) and characterize those representations satisfying σ(U)=R σ(U). We illustrate the theory developed so far by applications to representations on Banach lattices.

Quasi Constricted Linear Representations of Abelian Semigroups on Banach Spaces

Let (X, || · ||) be a Banach space. We study asymptotically bounded quasi constricted representations of an abelian semigroup ℙ in L(X), i.e. representations (Tt)t ∈ ℙ which satisfy the following conditions: i) for all x ∈ X. ii) X0 ≔ {x ∈ X : limt ∞||Ttx|| = 0} is closed and has finite codimension. We show that an asymptotically bounded representation (Tt)t ∈ ℙ is quasi constricted if and only if it has an attractor A with Hausdorff measure of noncompactness χ(A) < 1 with respect to some equivalent norm || · ||1 on X. Moreover we prove that every asymptotically weakly almost periodic quasi constricted representation (Tt)t ∈ ℙ is constricted, i.e. there exists a finite dimensional Tt)t ∈ ℙ-invariant subspace Xr such that X ≔ X0 ⊕ Xr. We apply our results to C0-semigroups.

Superstable semigroups of operators

Abstract In [NR] the authors introduced the notion of superstable operators on a Banach space E using ultrapowers Eu of E. In [HR] this notion was extended to strongly continuous one-parameter semigroups again by means of ultrapowers. It is the aim of the present paper to give an equivalent intrinsic definition of superstability (without the reference to ultrapowers). This definition allows us to improve the results of [NR] as well as of [HR]. We apply our results to semigroups of positive linear operators on Banach lattices and C ∗ - algebras , respectively.

On the Theory of Approximation by Positive Operators in Vector Lattices

Summary In this paper we give a survey and some new results in a field which can be described best by “theorems of Korovkin type in locally convex lattices”. The survey is given in part I. More precisely in § 1 besides notational preliminaries we give a short historical review. In § 2 we report on recent results which concern the universal Korovkin closure of subsets and universal Korovkin systems in general. § 3 establishes some very elementary facts concerning universal Korovkin systems. Kor example most of the known Korovkin systems are universal. This paragraph contains also an important characterization of vector lattices with finite Korovkin systems. Part II consists of new supplementary results with complete proofs. In § 4 we show to some extent the connection between Korovkin closures and general Choquet boundaries. The next paragraph is devoted to one main result stating that a sequentially complete locally convex lattice possesses a Korovkin system of 2 elements iff its dimension is at most 2. In the last paragraph we pose some open questions. The list of references is by no means complete. However, most of the papers refered to contain extensive references on their own, so that it is not difficult to obtain a complete survey of what has been done in the field.

Skew-product extensions of Markov operators and products of dependent random variables

In this paper we generalize the notion of skew products as known in ergodic theory to skew product extensions of Markov operators. We prove that Markov operators are of such a type iff they have relative discrete spectrum (in a slightly generalized sense) thus generalizing a theorem of Parry. In addition we show that skew product extensions of Markov operators play an important role in the theory of products of dependent random variables and we develop this interdependence between the two theories thus generalizing results of Koutsky, Schmetterer and Wolff.

Banach Spaces and Linear Operators

In this chapter we deal with old and new applications of nonstandard analysis to the theory of Banach spaces and linear operators. In particular we consider the structure theory of Banach spaces, basic operator theory, strongly continuous semigroups of operators, approximation theory of operators and their spectra, and the Fixed Point Property. To include in this chapter interesting examples of nonstandard functional analysis we must assume that the reader is familiar with the basics of Banach spaces and operator theory. Non-experts in these field can, however, profit from this chapter by looking at the elementary applications with which we begin every section. Moreover we refer to the book (Siu-Ah Ng, JAMA Nonstandard methods in functional analysis, 2010, [45]) for those, who want to learn functional analysis and simultaneously nonstandard analysis.

Analysis Alive—an Integrated Learning Environment for Higher Mathematics

We give an overview of Analysis Alive, a learning environment for Analysis at the level of a first year course at a German university. Analysis Alive is a traditional textbook in combination with a CD-ROM containing graphics and animations realized with Maple worksheets. The goals of our project are to introduce the component of experimentation with visual feedback into the teaching and learning of Analysis. Although Analysis Alive is based on the use of computers and Computer Algebra Systems, it requires only minimal knowledge of computers and computer algebra systems.