Vol. 17, No. 2, 1966

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On absolutely continuous functions and the well-bounded operator

William Hall Sills

Vol. 17 (1966), No. 2, 349–366

The author considers an operator T in a reflexive Banach space X for which there is a bounded operational calculus a a(T) defined on AC(I), the algebra of absolutely continuous functions defined on I = [0,1] with the norm |a(0)| + VarI(a) for a AC(I). Such operators, called well-bounded, have been investigated by Smart and Ringrose (J. Australian Math. Soc. 1 (1960), 319–343 and Proc. London Math. Soc. (3) 13 (1963), 613–638). The present paper explores a new method for obtaining the spectral theorem for this operator. Let AC0 be the maximal ideal of members of AC(I) which are zero at 0. The method consists in introducing Arens multiplication into AC0∗∗, the second conjugate space of AC0, and in investigating the larger algebra for a suitable family of idempotents which will serve as candidates for bounded spectral projections associated with T. Idempotents in AC0∗∗ are mapped into these projections by means of a homomorphism extension technique which extends the original operational calculus of AC0 into B(X) (the bounded linear operators on X), to a bounded homomorphism of AC0∗∗ into B(X). The extended homomorphism is defined on a quotient algebra of AC0∗∗. This quotient algebra turns out to be a copy of all functions of bounded variation on I which are zero at 0 under the usual pointwise operations.

Mathematical Subject Classification
Primary: 47.25
Received: 9 July 1964
Revised: 9 January 1965
Published: 1 May 1966
William Hall Sills