Hermitian and adjoint abelian operators on certain Banach spaces

Abstract

Let X be a complex linear space endowed with a semiinner product [ , ]. An operator A on X will be called Hermitian if [Ax, x] is real for all x 6 X; A is said to be adjoint abelian if [Ax, y] = [x, Ay] for all x and yeX. Since every Banach space may be given a semi-inner product (not necessarily unique) which is compatible with the norm, it is possible to study such operators on general Banach spaces. This paper characterizes Hermitian and adjoint abelian operators on certain Banach spaces which decompose as a direct sum of Hubert spaces. In particular, the Hermitian operators are shown to have operator matrix representations which are diagonal, with the operators on the diagonal being Hermitian operators on the appropriate Hubert space. The class of spaces studied includes those Banach spaces with hyperorthogonal Schauder bases.