A spectral duality theorem for closed operators (Q2266218)

The spectral duality theorem asserts that a densely defined closed operator T induces a spectral decomposition of the underlying Banach space X if and only if the conjugate \(T^*\) induces the same type of spectral decomposition of the dual space \(X^*\). This theorem is known for bounded linear operators in terms of residual (S-) decomposability. This paper extends the spectral duality theorem to unbounded closed operators endowed with the spectral decomposition property [Acta Sci. Math. 42, 67-70 (1980; Zbl 0436.47025)]. The proof of the ''if'' part of the theorem requires the domain-density assumptions on the first three successive conjugates \(T^*\), \(T^{**}\) and \(T^{***}\). For further results on this topic, see [the authors, ''A local spectral theory for closed operators'', London Math. Soc. Lecture Notes Series, 105].