An operator of the form Af = Gf' (first order) generates a semigroup of composition operators, for example on the Hardy space of the unit disk. Here the coefficient is a bounded holomorphic function.

The main part of the talk is devoted to operators of second order, given by

\[Af = (Gf')' + Hf' + Cf\]

with coefficients G, H, C which are functions on the unit disc. They generate holomorphic semigroups. A main tool for the proof is based on perturbation by weakly continuous forms, a notion we will discuss in the talk.

There is a surprise: For second order operators, the leading coefficient G may have a singularity at 0. A crucial argument uses a criterion for invariance of convex sets by a semigroup (in the case of second order) or a semiflow (in the first order case).

The talk is based on collaboration with I. Chalendar and B. Moletsane.

References:

W. Arendt, I. Chalendar, B. Moletsane: Perturbation by weakly continuous forms and semigroups on Hardy space.

J. Oper. Th. 86:2 (2021), 331-35.

W. Arendt, I. Chalendar: Generators of semigroups on Banach spaces inducing holomorphic semiflows.

Isr. J. Math. 229 (2019), 165-179.