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 ∈ X,A is said to be adjoint abelian if
[Ax,y] = [x,Ay] for all x and y ∈ X. 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 Hilbert 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 Hilbert space. The class of
spaces studied includes those Banach spaces with hyperorthogonal Schauder
bases.