Abstract
This note deals with a small but an important observation of hermitian operators on Banach spaces. It is known that if A is a complex Banach space, B(A) is the set of all operators on A, and H(B(A)) is the set of all hermitian operators, then B(A) = H(B(A)) + iH(B(A)) implies that A is a Hilbert space. We give a converse of this using norm one projections on A.
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