NORM ONE PROJECTIONS ON BANACH SPACES

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.

Mathematics Subject Classification (2000):

• 46H05
• 46J10
• 47B15
• 47B48

Key words:

• Projections
• ultra product of an operator
• hermitian operators
• direct sum of spaces