Abstract. In this paper we study a conjugation on a Banach space $\x $ and show properties of operators concerning conjugation $C$ and show spectral properties of such operators. Next we show spectral properties of an $(m,C)$-symmetry (isometry) operator $T$ on a complex Banach space $\x $. We prove that, for a $C$-doubly commuting pair $(T,S)$, if $T$ is an $(m,C)$-symmetry (isometry) and $S$ is an $(n,C)$-symmetry (isometry), then $T + S$ and $TS$ are $(m + n - 1,C)$-symmetries (isometries).
AMS Subject Classification
(1991): 47A05; 47B25, 47B99
Received June 4, 2018 and in final form August 31, 2018. (Registered under 51/2018.)