On spectral decomposition of closed operators on Banach spaces (Q1096873)

Some necessary and sufficient conditions for a closed operator T acting in a complex Banach space X to have the spectral decomposition property (SDP) are obtained. As the main result, the authors show that the following assertions are equivalent: (i) T has the SDP; (ii) T has property (k) and for every compact F in \(V_{\infty}\), \(\hat T=T/X(T,F)\) is closed and \(a(\hat T)\subset (IntF)^ c;\) (iii) For every relatively compact open G in the plane, there is an invariant subspace Y of T such that \(Y\subset D(T)\), \(\alpha(T| Y)\subset \bar G\), \(\hat T=T/Y\) is closed and \(\alpha(\hat T)\subset G^ c;\) (iv) T has property (k) and for every closed F in \(V_{\infty}\), \^T\(=T/X(T,F)\) is bounded and \(\alpha(\hat T)\subset (IntF)^ c;\) (v) For every relatively compact open G in the plane, there is an invariant subspace Y of T such that \(\hat T=T/Y\) is bounded, \(\alpha(T| Y)\subset G^ c\) and \(\alpha(\hat T)\subset \bar G\).