Break in Math by Peter Wolff

By Peter Wolff

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2 Hille-Yosida Theorems We discuss first what are probably the most well-known and perhaps most basic generation theorems that can be found in numerous other texts including [HP, Pa, Sh, T]. 1 (Hille-Yosida) For M ≥ 1, ω ∈ R, we have A ∈ G(M, ω) if and only if 1. , A closed and D(A) = X). 25 ✐ ✐ ✐ ✐ ✐ ✐ “K13799” — 2012/5/12 — 10:33 ✐ 26 ✐ A Functional Analysis Framework 2. For real λ > ω, we have λ ∈ ρ(A) and Rλ (A) satisfies |Rλ (A)n | ≤ M , (λ − ω)n n = 1, 2, . . 2 (Hille - Yosida) A ∈ G(1, 0) ⇐⇒ 1.

T (0) = I. (identity property) Classification of Semigroups by Continuity • T (t) is uniformly continuous if lim |T (t) − I| = 0. t→0+ This is not of interest to us, because T (t) is uniformly continuous if and only if T (t) = eAt where A is a bounded linear operator. • T (t) is strongly continuous, denoted C0 , if for each x ∈ X, t → T (t)x is continuous on [0, δ] for some positive δ. Note 1: All continuity statements are in terms of continuity from the right at zero. For fixed t T (t + h) − T (t) = T (t)[T (h) − T (0)] = T (t)[T (h) − I] and T (t) − T (t − ) = T (t − )[T ( ) − I] so that continuity from the right at zero is equivalent to continuity at any t for operators that are uniformly bounded on compact intervals.

If A ∈ G(1, ω), then A is densely defined, A − ωI is dissipative and R(λ0 − A) = X for all λ0 with Re λ0 > ω. 3 A is dissipative means Re Ax, x ≤ 0 Re −Ax, x ≥ 0. So we have: |(λ − A)x||x| ≥ ≥ ≥ = for all x ∈ D(A). This implies | (λ − A)x, x | Re (λ − A)x, x λ x, x λ|x|2 . Conversely, suppose |(λI − A)x| ≥ λ|x| for all x ∈ D(A) and λ > 0. Let x ∈ D(A). Define yλ = (λ − A)x and zλ = |yyλλ | . 1) ≤ λ|x||zλ | − Re Ax, zλ = λ|x| − Re Ax, zλ . This implies Re Ax, zλ ≤ 0. 2) We always have the relationship −Re Ax, zλ ≤ |Ax|.

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