Generalized 3-circular projections in some Banach spaces

TL;DR

This paper characterizes generalized n-circular projections in Banach spaces, showing they are sums of powers of a surjective isometry, extending known results for bi-circular projections.

Contribution

It generalizes the structure of n-circular projections in Banach spaces, providing explicit formulas involving surjective isometries for all n ≥ 3.

Findings

For n=3, projections are given by (I + T + T^2)/3 with T a surjective isometry.

For n > 3, the same form applies with T^n = I.

Results hold in classical Banach spaces like C(Ω) and Lp spaces.

Abstract

Recently in a series of papers it is observed that in many Banach spaces, which include classical spaces $C(\Omega)$ and $L_p$-spaces, $1 \leq p < \infty, p \neq 2$, any generalized bi-circular projection $P$ is given by $P = \frac{I+T}{2}$, where $I$ is the identity operator of the space and $T$ is a reflection, that is, $T$ is a surjective isometry with $T^2 = I$. For surjective isometries of order $n \geq 3$, 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 $P$, $n \geq 3$, is given by $P = \frac{I+T+T^2+...+T^{n-1}}{n}$ where $T$ is a surjective isometry and $T^n = I$. We prove our results for $n=3$ and for $n > 3$, the proof remains same except for routine modifications.