A systematic study is made of continuous linear operators approximable (in certain topologies) by linear combinations of projections from the range of a spectral set function. Such operators may be viewed as natural analogues of the scalar-type spectral operators introduced by N. Dunford. We extend the classical theory so that the range of the spectral set function, necessarily a Boolean algebra of continuous projection operators, need not be uniformly bounded; it is this feature which gives the theory its wider range of applicability. Typically, the associated operational calculus, which is specified via a suitable integration process, is no longer related to a continuous homomorphism but merely to a certain kind of sequentially closed homomorphism.
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Kluvánek, I., Ricker, W.J. Spectral set functions and scalar operators. Integr equ oper theory 17, 277–299 (1993). https://doi.org/10.1007/BF01200220
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