Extended Functional Calculus

Abstract

We extend our functional calculus for a contraction T on $\mathfrak{K}$ so that certain unbounded functions are also allowed. Let us recall the definitions of the classes $H^\infty_T$ and $K^\infty_T$ as given in Secs. 2 and 3 of the preceding chapter: $H^\infty_T$ consists of the functions $u\in H^\infty$ ??for which the strong operator limit $u(T)=lim_{r\rightarrow1-0} u_r(T)$ exists, and $K^\infty_T$ consists of those functions $u\in H^\infty_T$ for which $u(T)^{-1}$ exists and $K^\infty_T$ is densely defined in $\mathfrak{H}$. The class $H^\infty_T$ is an algebra, and the class $K^\infty_T$ is multiplicative.

For any operator T, the set Λ_T consists of the point λ = 0 and the symmetric image of ρ(T) ∖{0} with respect to the unit circle C, where ρ(T) denotes the resolvent set for TT. The set Λ_{T *} is the symmetric image of Λ_T with respect to the real axis.