Generalized 3-circular projections for unitary congruence invariant norms

Abstract

A projection $P_0$ on a complex Banach space is generalized $3$- circular if its linear combination with two projections $P_1$ and $P_2$ having coefficients ${\lambda }_1$ and ${\lambda }_2$, respectively, is a surjective isometry, where ${\lambda }_1$ and ${\lambda }_2$ are distinct unit modulus complex numbers different from $1$ and $P_0\oplus P_1\oplus P_2=I$. Such projections are always contractive. In this paper, we prove structure theorems for generalized $3$-circular projections acting on the spaces of all $n\times n$ symmetric and skew-symmetric matrices over $\mathbb{C}$ when these spaces are equipped with unitary congruence invariant norms.