Ergodic limits of tensor product semigroups

Abstract

Let e tS and e − tT be ( C 0)-semigroups on a Banach space X. Their tensor product L (t) is defined by L (t) A = e tSAe tT ( A ϵ B( X)) and has the generator Δ formally of the form ΔA = SA − AT. Under the assumption that { L ( t); t ⩾ 0} is bounded, we investigate the Abel limit and the Cesàro limit of L ( t) A at ∞. If gW s[Ω u] denotes the set of operators A for which the Abel limit P s ( A) [resp. P u ( A)] exists in the strong [resp. uniform] operator topology, then N(Δ)⊕ R(Δ) = ω u ⊂ ω s ⊂ N(Δ) + R(Δ) and the limit defines a projection P s [ P u ] from Ω s [resp. Ω u ] onto N( Δ) with N( Δ) with R(Δ) = N(P u) ⊂ N(P u) ⊂ R(Δ) . If, in addition, S and T are Hilbert space normal operators such that gq ( S) ∩ gq ( T) ≠ φ, then Ω u contains all compact operators.