THE SPECTRAL TYPE OF SUMS OF OPERATORS ON NON-HILBERTIAN BANACH LATTICES

Abstract

AbstractT. A. Gillespie showed that on a Hilbert space the sum of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. A longstanding question asked whether this might still hold true for operators on Lp spaces for $1\less p\less \infty $. We show here that this conjecture is false. Indeed for a large class of reflexive spaces, the above property characterizes Hilbert space.