By Klaus-Jochen Engel
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Extra resources for One-Parameter Semigroups for Linear Evolution Equations
The map K ×C (t, x) → F (t)x ∈ X is uniformly continuous for every compact set C in X. Proof. The implication (c) ⇒ (a) is trivial, while (a) ⇒ (b) follows from the uniform boundedness principle, since the mappings t → F (t)x are continuous, hence bounded, on the compact set K. To show (b) ⇒ (c), we assume F (t) ≤ M for all t ∈ K and ﬁx some ε > 0 and a compact set C in X. Then there exist ﬁnitely many x1 , . . , xm ∈ D m such that C ⊂ i=1 (xi + ε/M U ), where U denotes the unit ball of X. Now choose δ > 0 such that F (t)xi − F (s)xi ≤ ε for all i = 1, .
1) If A, B ∈ Mn (C) commute, then eA+B = eA eB . (2) Let A ∈ Mn (C) be an n×n matrix and denote by mA its minimal polynomial. , if the function (p−exp)/mA can be analytically extended to C, then p(A) = exp(A). 5). , etA ≤ M for all t ∈ R and some M ≥ 1, if and only if A is similar to a diagonal matrix with purely imaginary entries. (4) Characterize semigroups (etA )t≥0 satisfying eA = I in terms of the eigenvalues of the matrix A ∈ Mn (C). (5) A semigroup (etA )t≥0 for A ∈ Mn (C) is called hyperbolic if there exists a direct decomposition Cn = Xs ⊕ Xu into A-invariant subspaces Xs and Xu and constants M ≥ 1, ε > 0 such that etA x ≤ M e−εt x and etA y ≥ 1 εt e y M for all x ∈ Xs , t ≥ 0, for all y ∈ Xu , t ≥ 0.
Therefore, the original and the implemented automorphism groups coincide, and we can state the ﬁnal result. Theorem. Let H be a Hilbert space and take (U(t))t∈R to be a uniformly continuous group on L(H). Then the following properties are equivalent. (a) (U(t))t∈R is a group of ∗ -automorphisms on the Banach∗ -algebra L(H). (b) There exists a skew-adjoint operator A ∈ L(H) and a unitary group (etA )t∈R on H such that ∗ U(t)T = etA · T · etA for all T ∈ L(H). 17 Exercises. (1) On the Banach space X := C0 (R) and for a ﬁxed constant α > 0, we deﬁne an operator Aα by the diﬀerence quotients Aα f (s) := 1/α f (s + α) − f (s) , f ∈ X, s ∈ R.
One-Parameter Semigroups for Linear Evolution Equations by Klaus-Jochen Engel