Intermediate Spaces and Semi-Groups

Abstract

In Chapter II we discussed necessary and sufficient condition supon an element f ∈ X such that || [T (t) — I]^r f || (r any fixed positive integer) is precisely of optimal order O(t^r) (t → 0+) — the saturation problem (Theorem 2.2.14) — as well as conditions such that || [T (t) — I]^r f || is of order O(t^α) (0 < α < r, t → 0+) in the special case r = 1 — the problem of non-optimal approximation (Corollary 2.3.6) — where {T(t); 0 ≦ t < ∞} is a strongly continuous semi-group of operators on the Banach space X.