HERMITIAN OPERATORS ON BANACH ALGEBRAS OF VECTOR-VALUED LIPSCHITZ MAPS (Researches on isometries from various viewpoints)

Abstract

Let H be a complex Hilbert space and [., .] an inner-product on H. A bounded linear operator T on H is a Hermitian operator if [Tx, x] in mathbb{R} for each x in H. In 1961, the Hermitian operator on a normed vector space was defined by means of the semi-inner product defined by Lumer [6]. Hermitian operators and their applications have been studied by many authors; a few of them are [1, 2, 5, 6, 7]. We exhibit forms of Hermitian operators on certain semisimple commutative Banach algebras.