By Ol'shanskii A. Y.

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**Example text**

We introduce the space D(A∞ ) = D(An ). 8. 6. If A is the generator of a strongly continuous semigroup on X, then D(A∞ ) is dense in X. Proof. We denote by D(0, 1) the space of all inﬁnitely diﬀerentiable functions on (0, 1) whose support is compact and contained in (0, 1). We denote by T the semigroup generated by A. For every ϕ ∈ D(0, 1) we deﬁne the operator Tϕ by 1 Tϕ z0 = ϕ(t)Tt z0 dt 0 ∀ z0 ∈ X . 3. The resolvents of a semigroup generator 31 Take z0 ∈ D(A). 5 that the integral in the deﬁnition of Tϕ z0 may be considered as an integral in D(A) (with the graph norm) and Tϕ z0 ∈ D(A).

14) iﬀ z is continuous with values in D(A) (endowed with the graph norm), continuously diﬀerentiable with values in X and it satisﬁes the equations ∀t z(t) ˙ = Az(t) 0 , z(0) = w0 . • The family of functions (ϕk )k∈N , deﬁned by ϕk (x) = 2 cos π k− 1 2 ∀ k ∈ N, x ∈ (0, π), x consists of eigenvectors of A, it is an orthonormal basis in X and the corresponding eigenvalues are λk = − k − 1 2 2 ∀ k ∈ N. • 0 ∈ ρ(A). 5, A is the generator of a strongly continuous semigroup T on X given by 1 2 e−(k− 2 ) Tt z = t z, ϕk ϕk ∀t 0, z ∈ X .

Strongly continuous groups 49 U ∗ ). A strongly continuous semigroup T on X is called unitary if Tt is unitary for every t > 0. It is clear that a unitary semigroup can be extended to a group, which is then called a unitary group. 8 we shall give a simple characterization of the generators of unitary groups (the theorem of Stone). Three simple examples of unitary groups will be given in this section. 10. If there is in X an orthonormal basis formed by eigenvectors of A, then T is unitary iﬀ Re λ = 0 for all λ ∈ σ(A).