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The spectral decomposition property has been instrumental in developing a local spectral theory for closed operators acting on a complex Banach space. This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property. In the monograph [3] and in a sequel of papers by the authors, a local spectral theory has been built for closed operators on the sole assumption of the spectral decomposition property. As an abstraction of Dunford's concept of "spectral reduction' [2, p. 1927] and that of Bishop's "duality theory of type 3" [1], an operator T endowed with the spectral decomposition property produces a spectral decomposition of the underlying space, pertinent to any finite open cover of the spectrum a( T) . In this paper we obtain some conditions equivalent to the spectral decomposition property. Some of them generalize results from [4].

Equivalent conditions to the spectral decomposition property for closed operators