# Complex symmetry of first-order differential operators on Hardy space

## TL;DR

This paper characterizes complex symmetric and hermitian first-order differential operators on Hardy space, providing conditions for symmetry, comparing classes, and analyzing their spectra.

## Contribution

It introduces a characterization of complex symmetric differential operators on Hardy space and compares them with hermitian operators, including spectral analysis.

## Key findings

- Hermitian operators are a proper subset of $\\calC$-selfadjoint operators.
- Explicit spectral calculations for certain operators.
- Complete characterization of complex symmetry conditions.

## Abstract

Given holomorphic functions $\psi_0$ and $\psi_1$, we consider first-order differential operators acting on Hardy space, generated by the formal differential expression $E(\psi_0,\psi_1)f(z)=\psi_0(z)f(z)+\psi_1(z)f'(z)$. We characterize these operators which are complex symmetric with respect to weighted composition conjugations. In parallel, as a basis of comparison, a characterization for differential operators which are hermitian is carried out. Especially, it is shown that hermitian differential operators are contained properly in the class of $\calC$-selfadjoint differential operators. The calculation of the point spectrum of some of these operators is performed in detail.