Spectral theory and functional calculus for operators on spaces of generalized functions (Q1191852)

Suppose \(T\) is a densely defined operator on a complex separable Hilbert space \(H\). Then \(D_ \infty= \bigcap_{n=1}^ \infty D(T^ n)\) becomes a Fréchet space in the topology determined by the seminorms \(\| T^ n x\|\). This space is dense in \(H\) and its dual \(D_ \infty'\) is sequentially complete. The extension \(\overline {T}\) of \(T\) on \(D_ \infty'\) is defined by the relation \(\langle \overline{T}y,x\rangle= \langle y,Tx\rangle\), \(\forall x\in D_ \infty\), \(y\in D_ \infty'\) i.e. this is the \(D_ \infty\) adjoint of \(T\). Now let \(E(\lambda)\) be the spectral function of \(T\). For every Borel measurable function \(F\) on \(\mathbb{R}\), the operator \(F(T)\) is defined by \(F(T)x=\int_ R F(\lambda)dE(\lambda)x\). The main theorem in the paper describes all such functions which give rise to continuous operators on \(D_ \infty\). Namely, \(F(T)\) is continuous on \(D_ \infty\), iff \(\sup_{\lambda\in\sigma(T)} | F(\lambda)| (1+\lambda^ 2)^{- m}= M<\infty\), for some \(m>0\). By definition, \(\overline {F(T)}= F(\overline{T})\). The authors show that \(F\to \overline{F(T)}\in L(D_ \infty)\) is a continuous functional calculus. They also provide an interesting application: starting with \(F_ t(\lambda)=e^{t\lambda}\), \(t>0\), they obtain a semigroup of operators on \(D_ \infty\) which is the solution of a weak abstract Cauchy problem.