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Register a new userOn the numerical range of generators of symmetric $L_infty$-contractive semigroups
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A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Krieglers result.
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The numerical range of a bounded, linear operator on a Hilbert space is a set in $mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 ominus theta H^2$ where $theta$ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture.
We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_infty$ operator norm of the semigroup. As an application we prove a new $L_infty$-bound for the torsion function of a Euclidean domain that is close to optimal.
Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used also in the unbounded case, but rather little is known about the general form of unbounded generators. In this paper we first give a precise description of the standard form in the unbounded case, emphasizing intuition, and collecting and even proving the basic results around it. We also give a cautionary example showing that the standard form must not be read too naively. Further examples are given of semigroups, which appear to be probability preserving to first order, but are not for finite times. Based on these, we construct examples of generators which are not of standard form.
Let $mathfrak{n}$ be a nonempty, proper, convex subset of $mathbb{C}$. The $mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.
We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contactive idempotent.
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