# Complex symmetric evolution equations

## TL;DR

This paper investigates complex symmetric evolution equations, characterizing generators of semigroups and their properties in bounded and unbounded operator frameworks, with applications to Fock space and weighted composition semigroups.

## Contribution

It provides new characterizations of complex symmetric semigroups, including conditions for generators and their properties, extending the theory to unbounded operators and specific realizations.

## Key findings

- A $ ext{C}$-selfadjoint operator generates a contraction $C_0$-semigroup iff it is dissipative.
- Characterization of $ ext{C}$-selfadjoint, unbounded weighted composition semigroups on Fock space.
- The generator of a $ ext{C}$-selfadjoint, unbounded semigroup need not be $ ext{C}$-selfadjoint.

## Abstract

We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a $\mathcal{C}$-selfadjoint operator generates a contraction $C_0$-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of \emph{complex symmetric, unbounded semigroups} and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all $\mathcal{C}$-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a $\mathcal{C}$-selfadjoint, unbounded semigroup is not necessarily $\mathcal{C}$-selfadjoint.