Jochen Glück

Eventually positive semigroups of linear operators

Abstract We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron–Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation. We also consider eventually positive semigroups on arbitrary Banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups.

Eventually and asymptotically positive semigroups on Banach lattices

Abstract We develop a theory of eventually positive C 0 -semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on L p -spaces, the Dirichlet-to-Neumann operator on L 2 and the Laplacian with non-local boundary conditions on L 2 within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call “asymptotic positivity”, where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as t → ∞ . This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.

On the peripheral spectrum of positive operators

This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive operator is cyclic and that the corresponding eigenspaces fulfil a certain dimension estimate. A couple of examples demonstrates that some of our theorems are optimal. Our results on the peripheral point spectrum are then used to prove a sufficient condition for the peripheral spectrum of a positive operator to be cyclic; this generalizes theorems of Lotz and Scheffold.

Invariant sets and long time behaviour of operator semigroups

Institute und Sektionen der Fakultäten für Mathematik und Wirtschaftswissenschaften Naturwissenschaften Bitte um Bekanntgabe an das wissenschaftliche Personal und die Doktoranden Sowie an den Präsidenten, die Vizepräsidenten, die Dekane der Medizinischen Fakultät und der Fakultät f. Ingenieurwissenschaften und Informatik mit der Bitte um Aushang Promotionssekretariat für den Dr. rer. nat. Neben der Promotionskommission sind alle oben eingeladenen Professoren, Hochschul-und Privatdozenten frageberechtigt. Die mündliche Promotionsprüfung (ausschließlich der Beratung und Bekanntgabe der Prüfungsergebnisse) ist nach Maßgaben der vorhandenen Plätze öffentlich.

The role of domination and smoothing conditions in the theory of eventually positive semigroups

We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron--Frobenius type spectral theorems. We furthermore prove a Kre\uin--Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

On weak decay rates and uniform stability of bounded linear operators

We consider a bounded linear operator T on a complex Banach space X and show that its spectral radius r(T ) satisfies r(T ) < 1 if all sequences (〈x, Tnx〉)n∈N0 (x ∈ X, x ′ ∈ X) are, up to a certain rearrangement, contained in a principal ideal of the space c0 of sequences which converge to 0. From this result we obtain generalizations of theorems of G. Weiss and J. van Neerven. We also prove a related result on C0-semigroups.

A toolkit for constructing dilations on Banach spaces: A TOOLKIT FOR CONSTRUCTING DILATIONS ON BANACH SPACES

We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if

A note on lattice ordered C∗-algebras and Perron-Frobenius theory

X

A Criterion for the Uniform Eventual Positivity of Operator Semigroups

is a super-reflexive Banach space and

A note on approximation of operator semigroups

T