Interpolation theorems for conjugations and applications

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.