On similarity to contraction semigroups and tensor products, II: Infinite tensor products

Abstract

We develop a framework for infinite tensor products of Hilbert spaces, operators, and semigroups tailored to questions of similarity to contraction semigroups. On the operator-theoretic side, we give a systematic treatment of incomplete infinite tensor products, including criteria for existence, non-vanishing, and continuity properties of the associated tensor product semigroups. On the semigroup-theoretic side, we prove a low-regularity similarity theorem showing that global quasi-contractive control together with local contractive information at one positive time implies similarity to a contraction semigroup, with explicit bounds on the similarity constant. These ingredients are then combined to obtain infinite analogues of the finite tensor-product splitting principle for similarity to contraction and quasi-contraction semigroups. We also clarify the role of complete tensor products and show, in particular, that whenever a complete infinite tensor product of semigroups is a \(C_0\)-semigroup, it decomposes along incomplete tensor-product components.