Generalized Daugavet Equations and Invertible Operators on Uniformly Convex Banach Spaces

✍ Scribed by C.-S. Lin

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 156 KB
• Volume: 197
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1996.0036

✦ Synopsis

In this article we shall introduce and investigate a notion of generalized 5 5 5 5 5 5 Daugavet equation I q S q T s 1 q S q T for operators S and T on a uniformly convex Banach space into itself, where I denotes the identity operator. This extends the well-known Daugavet equation 5 5 5 5 IqT s1q T . The operators satisfying this equation will be characterized in terms of their approximate point spectrum. From this it is shown 5 5 in particular that if I y rT s 1 for some r ) 1, then T is invertible if 5 5 and only if I y T -1. A characterization of a noninvertible operator in terms of the Daugavet equation is also given. The equation for operators on a Hilbert space is investigated, and these results make it possible to characterize a normal approximate proper value of an operator.

Throughout this paper, unless otherwise stated explicitly, we shall as-Ž . sume that X is a uniformly convex Banach space and B X is the Banach algebra of all bounded linear operators on X. Recall that X is uniformly Ä 4 Ä 4 5 5 5 5 convex if for any sequences x and y satisfying x F 1 and y F 1