Abstract and Applied Analysis
Volume 3 (1998), Issue 3-4, Pages 237-246
doi:10.1155/S1085337598000542
Spectral properties of operators that characterize ℓ∞(n)
B.L. Chalmers1
and B. Shekhtman2
1Department of Mathematics, University of California, Riverside 92521, California, USA
2Department of Mathematics, University of South Florida, Tampa 33620-5700, Florida, USA
Abstract
It is well known that the identity is an operator with the following property: if the operator, initially defined on an n-dimensional Banach space V, can be extended to any Banach space with norm 1, then V is isometric to ℓ∞(n). We show that the set of all such operators consists precisely of those with spectrum lying in the unit circle. This result answers a question raised in [5] for complex spaces.