Algebraic and invariance properties of the group of isometries

In this paper we investigate the algebraic structure of the isometry group of several classical Banach spaces, namely, symmetric sequence spaces, Cp(H)Cp(H) (1≤p<∞,p≠2), L(ℓp,ℓr)L(ℓp,ℓr), 1≤p,r<∞1≤p,r<∞, and bounded operators L(H)L(H) endowed with Chan–Li–Tui unitarily invariant (c,p)(c,p)-norms. We also identify the isometrically invariant subspaces for each of theses settings.