Abstract
Let X be a complex linear space endowed with a semi-inner product [ , ]. An operator A on X will be calledHermitian if [Ax, x] is real for all x 6 X; A is said to beadjoint abelian if [Ax, y] = [x, Ay] for all x and yeX. Sinceevery Banach space may be given a semi-inner product (notnecessarily unique) which is compatible with the norm, it ispossible to study such operators on general Banach spaces.This paper characterizes Hermitian and adjoint abelian opera-tors on certain Banach spaces which decompose as a directsum of Hubert spaces. In particular, the Hermitian operatorsare shown to have operator matrix representations which arediagonal, with the operators on the diagonal being Hermitianoperators on the appropriate Hubert space. The class of spacesstudied includes those Banach spaces with hyperorthogonalSchauder bases.
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