On the joint spectra of commuting tuples of operators and a conjugation

Abstract

In this paper we show that if ${\bf T} = (T_1,...,T_n)$ is a commuting $n$-tuple of Hilbert space operators and $C$ is a conjugation, then $\sigma(C{\bf T}C) = \sigma({\bf T})^*$, where $\sigma(C{\bf T}C) = (CT_1C,...,CT_nC)$, $\sigma({\bf T})$ is the Taylor spectrum of ${\bf T}$ and $\sigma({\bf T})^* = \{ \, \overline{z} = (\overline{z_1},...,\overline{z_n})  \, : \, z=(z_1,...,z_n) \in \sigma({\bf T}) \}.$   We characterize joint approximate point spectra of  $m$-symmetric tuples, $m$-complex symmetric tuples, skew $m$-complex symmetric tuples,  $[m,C]$-symmetric tuples and skew $[m,C]$-symmetric tuples.