A Bounded Compact Semigroup on Hilbert Space not Similar to a Contraction One

Abstract

Let H be a Hilbert space and let B (H ) be the Banach space of all bounded operators on H. By definition we say that T ∈ B (H) is similar to a contraction if there exists an invertible operator S ∈ B(H) such that ║ S ^-1 TS ║ ≤ 1. Likewise, we say that a c_o-semigroup (T _t ) _t≥0on H is similar to a contraction semi-group if there exists an invertible operator S ∈ B (H) such that ║ S ^-1 T _t S ║ ≤ 1 for any t ≥ 0. Clearly any T ∈ B(H) similar to a contraction is power bounded, i.e., sup {║ T ⁿ║ : n ∈ IN} < ∞, and any c₀-semigroup (T _t ) _t≥0similar to a contraction one is bounded, i.e., sup {║ T _t ║: t ≥ 0} < ∞. However it has been known for a long time that these conditions are not sufficient to ensure similarity to contraction ([F], [P]). Besides some deep research made to characterize power bounded operators similar to contraction (see [Pa, Bo, Pil, Pi2]), a recent attempt was made by Q.-P. Vu and F. Yao ([VY]) and by the author (L]) to classify bounded co-semigroups similar to contraction ones. The situations for single operators and for semigroups are quite different. For example, it was noticed both in [VY] and [L] that the property that lim_t→ ∞ ║ T _t║ = 0 does not imply that (T _t) _t≥0is similar to a contraction semigroup, whereas any T ∈ B(H) such that lim_{n→ ∞} ║ T ⁿ║ = 0 is similar to a contraction (Rota’s Theorem).