Some types of Banach spaces, Hermitian operators, and Bade functionals
Earl Berkson · Transactions of the American Mathematical Society · 1965
Introduction.In [6] and [ll] a general notion of hermitian operator has been developed for arbitrary complex Banach spaces (see §1 below).In terms of this notion, a family of operators on a Banach space is said to be hermitian-equivalent if the operators of this family can be made simultaneously hermitian by equivalent renorming of the underlying space [7].Let X be a complex Banach space with norm || ||, and let F be a commutative hermitian-equivalent family of operators on X.A norm for X equivalent to || ||, and relative to which the operators of F are hermitian will be called an F-norm.Such families have been studied in [7], where it is shown that if X is a Hilbert space, then there is an F-norm which is also a Hilbert space norm.It is natural to seek other properties which, if enjoyed by X, can be preserved by choosing an F-norm appropriately.Such an investigation is conducted in this paper.Specifically, we show in § §3 and 4 that if the Banach space X has uniformly Frechet differentiable norm (resp., is uniformly convex), then there is an F-norm which preserves uniform Frechet differentiability (resp., uniform convexity).Moreover, we show in §6 that if X = Lp(fi), > p > 1, M a measure, then there is an F-norm which preserves both uniform Frechet differentiability and uniform convexity.Our result for'the case where X is uniformly convex enables us to establish in Theorem (5.4) a strong link between the notions of semi-innerproduct (see §1) and Bade functional.This link adds to the analogy with Hilbert space inherent in these notions.Throughout this paper all spaces are over the complex field, and an operator will be a bounded linear transformation with range contained in its domain.In some cases it will be convenient to employ a notation for Banach spaces which explicitly exhibits the norm.Thus, if Y is a linear space, and | | is a Banach space norm for Y, we shall sometimes designate the resulting Banach space by (Y,| |).1. Semi-inner-products and Hermitian operators.In this section and the next we reproduce some machinery from other papers which will be needed