Interpolation theorems for conjugations and applications

TL;DR

This paper investigates conditions for the existence of conjugations on a Hilbert space that map given orthonormal sets and relate to normal operators, with applications to complex symmetric operators and hyperinvariant subspaces.

Contribution

It provides complete spectral conditions for constructing conjugations satisfying specific intertwining properties with normal operators.

Findings

Characterizes when conjugations exist for given orthonormal sets and normal operators.

Connects conjugation existence to spectral projections of normal operators.

Applies results to complex symmetric operators and hyperinvariant subspace theory.

Abstract

Let $\mathcal{H}$ be a separable complex Hilbert space. A conjugate-linear map $C:\mathcal{H}\to \mathcal{H}$ is called a conjugation if it is an involutive isometry. In this paper, we focus on the following interpolation problems: Let $\{x_i\}_{i\in I}$ and $\{y_i\}_{i\in I}$ be orthonormal sets of vectors in $\mathcal{H}$, and let $\{N_k\}_{k\in K}$ be a set of mutually commuting normal operators. We seek to determine under which conditions there exists a conjugation $C$ on $\mathcal{H}$ such that \begin{enumerate}[\rm (a)] \item $Cx_i=y_i$ and $CN_kC=N_k^*$ for all $i\in I$ and $k\in K$; or \item $Cx_i=y_i$ and $CN_kC=-N_k^*$ for all $i\in I$ and $k\in K$. \end{enumerate} We provide complete answers to problems (a) and (b) using the spectral projections of normal operators. Our results are then applied to the study of complex symmetric and skew symmetric operators, as well as to the characterization of hyperinvariant subspaces of normal operators through conjugations.