An Agler-type model theorem forC₀-semigroups of Hilbert space contractions

Abstract

We investigate suitable conditions for a C0‐semigroup false(T(t)false)t⩾0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup false(Sγ*(t)false)t⩾0 to an invariant subspace of the standard weighted Bergman space Aγ−2false(C+,scriptKfalse). It turns out that false(T(t)false)t⩾0 admits a model by false(Sγ*(t)false)t⩾0 if and only if its cogenerator is γ‐hypercontractive and trueprefixlimt→0Tfalse(tfalse)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ‐hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization.

$$\varvec{\beta }$$ β -admissibility of observation operators for hypercontractive semigroups

“…As in [21], the cogenerator T := (A + I)(A − I) −1 exists, and is itself a contraction. Rydhe [20] studied the relation between γ-hypercontractivity of a strongly continuous contraction semigroup and its cogenerator. He proved that T is γ-hypercontractive if every operator T (t), t ≥ 0, is γ-hypercontractive.…”

“…Conversely, if every operator T (t), t ≥ 0, is N -hypercontractive for some N ∈ N, then T is N -hypercontractive. However, by means of an example, Rydhe [20] showed that for general γ-hypercontractivity this reverse implication is false. Clearly, if A generates a contraction semigroup of normal operators, then the cogenerator of (T (t)) t≥0 is γ-hypercontractive for each γ ≥ 1.…”

“…In particular 2-hypercontractivity can be characterized as follows, see [20]. For completeness we include a more elementary proof, which also yields additional information.…”

Some Examples of m-Isometries

“…Another way to construct examples of m-isometries is developing different tools like tensor product [19], functional calculus [24], on Hilbert-Schmidt class [17] and with C 0 -semigroups [10,21,29].…”