Generalized 3-circular projections in some Banach spaces

Recently in a series of papers it is observed that in many Banach spaces, which include classical spaces $C(\mathrm{\Omega })$ and $L_p$-spaces, $1 \leq p < \infty, p \neq 2$,anygeneralizedbi-circularprojection$P$isgivenby$P = \frac{I+T}{2}$,where$I$istheidentityoperatorofthespaceand$T$ is a reflection, that is, $T$ is a surjective isometry with $T^2=I$. For surjective isometries of order $n\ge 3$, the corresponding notion of projection is generalized $n$-circular projection as defined in \cite{AD}. In this paper we show that in a Banach space $X$, if generalized bi-circular projections are given by $\frac{I+T}{2}$ where $T$ is a reflection, then any generalized $n$-circular projection $P$, $n\ge 3$, is given by $P = \frac{I+T+T^2+…+T^{n-1}}{n}$where$T$isasurjectiveisometryand$T^n = I$\mathrm{.}Weproveourresultsfor$n=3$andfor$n > 3$, the proof remains same except for routine modifications.