In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

T. 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 . We show here that this conjecture is false. Indeed for a large class of reflexive spaces, the above property characterizes Hilbert space.

Motivated by problems in the spectral theory of linear operators, we previously introduced a new concept of variation for functions defined on a non-empty compact subset of the plane. In this paper, we examine the algebras of functions of bounded variation one obtains from these new definitions for the case where the underlying compact set is either a rectangle or the unit circle, and compare these algebras with those previously used by Berkson and Gillespie in their theories of AC-operators and trigonometrically well-bounded operators.