Linear operators and conjugations on a Banach space

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).