Dilations of markovian semigroups of measurable Schur multipliers

TL;DR

This paper proves that certain semigroups of Schur multipliers can be dilated into groups of automorphisms, leading to new boundedness results and answering open questions in operator theory.

Contribution

It introduces a dilation technique for weak* continuous semigroups of Schur multipliers, extending the theory of operator dilations and functional calculus.

Findings

Established dilation of semigroups to automorphism groups

Proved boundedness of McIntosh's $ ext{H}^ty$ calculus on Schatten spaces

Provided an answer to a question on completely positive Schur multipliers

Abstract

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm{B}(\mathrm{L}^2(X))$ of bounded operators on the Hilbert space $\mathrm{L}^2(X)$, where $X$ is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to $\mathrm{BMO}$-spaces. We also give an answer to a question of Steen, Todorov and Turowska on completely positive continuous Schur multipliers.