Address: (FL & GL) Laboratoire de Mathématiques, UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France
Abstract: It is known that on a Hilbert space, the sum of a real scalar-type operator and a commuting well-bounded operator is well-bounded. The corresponding property has been shown to be fail on Lp spaces, for 1 < p ‡ 2 < \infty. We show that it does hold however on every Banach space X such that X or X* is a Grothendieck space. This class notably includes L1 and C(K) spaces.
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