Hermitian operators and one-parameter groups of isometries in Hardy spaces

Abstract:

Call an operator A with domain and range in a complex Banach space X hermitian if and only if iA generates a strongly continuous one-parameter group of isometries on X. Hermitian operators in the Hardy spaces of the disc $({H^p},1 \leq p \leq \infty )$ are investigated, and the following results, in particular, are obtained. For $1 \leq p \leq \infty ,p \ne 2$, the bounded hermitian operators on ${H^p}$ are precisely the trivial ones—i.e., the real scalar multiples of the identity operator. Furthermore, as pointed out to the authors by L. A. Rubel, there are no unbounded hermitian operators in ${H^\infty }$. To each unbounded hermitian operator in the space ${H^p},1 \leq p < \infty ,p \ne 2$, there corresponds a uniquely determined one-parameter group of conformal maps of the open unit disc onto itself. Such unbounded operators are classified into three mutually exclusive types, an operator’s type depending on the nature of the common fixed points of the associated group of conformal maps. The hermitian operators falling into the classification termed “type (i)” have compact resolvent function and one-dimensional eigenmanifolds which collectively span a dense linear manifold in ${H^p}$.