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Form Inequalities for Symmetric Contraction Semigroups

  • Markus HaaseEmail author
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 255)

Abstract

Consider – for the generator –A of a symmetric contraction semigroup over some measure space \(X, 1\leq p< \infty\), q the dual exponent and given measurable functions \(F_j,\;G_j\;:\;\mathbb{C}^{d}\;\rightarrow\;\mathbb{C}\) – the statement:
$$Re\sum_{j=1}^{m}\int_X AF_{j}(f)\cdot G_{j}(f)\;\geq\;0$$
for all \(\mathbb{C}^{d}\) -valued measurable functions f on X such that \(F_{j}(f)\;\in\;\mathrm{dom}(A_{p})\) and \(G_{j}(f)\;\in\;L^{q}(X)\) for all j.

It is shown that this statement is valid in general if it is valid for X being a two-point Bernoulli \(\begin{array}{lll}\big(\frac{1}{2},\frac{1}{2}\big)\end{array}\)-space and A being of a special form. As a consequence we obtain a new proof for the optimal angle of Lp-analyticity for such semigroups, which is essentially the same as in the well-known sub- Markovian case.

The proof of the main theorem is a combination of well-known reduction techniques and some representation results about operators on C(K)-spaces. One focus of the paper lies on presenting these auxiliary techniques and results in great detail.

Keywords

Symmetric contraction semigroup diffusion semigroup sector of analyticity Stone model integral bilinear forms tensor products 

Mathematics Subject Classification (2010).

47A60 47D06 47D07 47A07 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Christian-Albrechts Universität zu KielMathematisches SeminarKielGermany

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