Triviality of the Peripheral Point Spectrum

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 Schef…

A note on the spectrum of irreducible operators and semigroups

Let $T$ denote a positive operator with spectral radius $1$ on, say, an $L^p$-space. A classical result in infinite dimensional Perron--Frobenius theory says that, if $T$ is irreducible and power bounded, then its peripheral point spectrum is either empty or a subgroup of the unit circle. In this note we show that the analogous assertion for the entire peripheral spectrum fails. More precisely, for every finite union $U$ of finite subgroups of the unit circle we construct an irreducible stochastic operator on $\ell^1$ whose peripheral spectrum equals $U$. We also give a similar construction fo…

Hilbert与Banach空间中算子变换与谱投影的超限迭代

Transfinite Iteration of Operator Transforms and Spectral Projections in Hilbert and Banach Spaces

研究了有界算子变换多层迭代的收敛性，在泛函演算假设下证明了向谱投影的强收敛，并提供了显式滤波器、单调性界及反例。

Spectral properties of locally eventually positive operator semigroups

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral po…

The Wavelet Galerkin Operator

In connection to wavelet theory, we describe the peripheral spectrum of the transfer operator. The solution involves the analysis of certain representations of the algebra generated by two unitaries $U$ and $T$ that satisfy the commutation relation $UTU^{-1}=T^N$.

Classes of contractions and Harnack domination

Several properties of the Harnack domination of linear operators acting on Hilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g. convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it.