SOME INVARIANT SUBSPACES FOR BOUNDED LINEAR OPERATORS

A bounded linear operator T on a complex Banach space Xis said to have property (I) provided that T has Bishop’s property () andthere exists an integer p > 0 such that for a closed subset F of C where XT (F) denote the analytic spectral subspace and ET (F) denote the algebraic spectral subspace of T: Easy examples are provided by normal operators and hyponormal operators in Hilbert spaces, and more generally, generalized scalar operators and subscalar operators in Banach spaces. In this paper, we prove that if T has property (I); then the quasi-nilpotent part H0(T) of T is given by for all sufficiently large integers p; where rT (x) = lim supn→∞ jjTnxjj 1n : We also prove that if T has property (I) and the spectrum (T) is finite, then T is algebraic. Finally, we prove that if T 2 L(X) has property (I) and has decomposition property (); then T has a non-trivial invariant closed linear subspace.