Isometries and isometric equivalence of hermitian operators on A1,p(X)

Let X be a separable complex Banach space with no nontrivial L1-projections and not isometrically isomorphic to Lp([0, 1],X), where 1 < p < ∞, p = 2. The space A1,p(X) is defined to be the set of all absolutely continuous functions f : [0, 1] → X such that df dx exist a.e. on (0, 1) and belongs to Lp([0, 1],X). If f ∈ A1,p(X), the norm of f on this space is defined to be |||f ||| = f (0) X + f Lp([0,1],X). We prove that if T is a surjective isometry T of A1,p(X), then T is given by Tf (x) = T0f (0)+ x 0 U(f )(t) dt, where T0 is a surjective isometry of X and U is a surjective isometry of Lp([0, 1],X). We also give the form of a hermitian operators on A1,p(X). In addition, if we assume that X is not the lp-direct sum of two nonzero Banach spaces (for the same p), we obtain the conditions of isometric equivalence of two hermitian operators on A1,p(X). © 2007 Elsevier Inc. All rights reserved