Growth conditions and the numerical range in a Banach algebra
Joseph G. Stampfli, J. P. Williams · Tohoku Mathematical Journal · 1968
Introduction.In this paper, we consider the numerical range in an arbitrary Banach algebra with identity, and study its relation to various growth conditions on the resolvent.In §1, we list several facts about this generalized numerical range.Some of these are more or less well known for concrete algebras, but do not seem to have been formulated in their proper generality.In particular, we recognize that part of this section is implicit or explicit in Lumer [7].Consequently, we have included proofs here only when a result is new (or requires a new technique), or when the proof represents a substantial simplification.The main result of §2 is a Phragmen-Lindelδf theorem for quasi-nilpotent elements in a Banach algebra.This yields a sharper version of a similar theorem of Lumer and Phillips [8].Finally, in §3 we apply the methods of §1 to study the numerical range in its usual setting, Hubert space.The notion of essential numerical range appears naturally here, and this set is shown to be chracterized in the way one would expect by analogy with the essential spectrum.