Vol. 52, No. 1, 1974

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Hermitian and adjoint abelian operators on certain Banach spaces

Richard Joseph Fleming and James E. Jamison

Vol. 52 (1974), No. 1, 67–84

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.

Mathematical Subject Classification 2000
Primary: 47B15
Received: 9 February 1973
Revised: 14 June 1973
Published: 1 May 1974
Richard Joseph Fleming
James E. Jamison