On Characterizations of Isometries on Function Spaces

Let X be a Banach space over C. Let T : X → X be an isometric isomorphism (an onto linear operator on X with |Tx| = |x| for all x ∈ X). When X is a Hilbert space, any perturbations of orthonormal basis of X give an isometric isomorphism on X. But, for non-Hilbertian space, one expects that an isometric isomorphism should be something special, and would like to find a characterization for T on some Banach spaces X like L space, Hardy space H when p 6= 2, etc.. Questions along this line have attracted a lot attention. For D being a bounded domain in C, we let H(D) be the holomorphic Hardy space over D, and let A(D) be the Bergman space over D. For any holomorphic map φ : D → D, we let Cφ be the composition operator associated to φ acting on function spaces over D, namely, Cφ(u) = u ◦φ for any function u on D. For 1 ≤ p ≤ ∞ and p 6= 2, we let T : H(D) → H(D) be an isometric isomorphism. It has been proved by several authors for certain class of domains D that