ON INVERTIBILITY IN NONSELF-ADJOINT OPERATOR-ALGEBRAS

Let L be a complete commutative subspace lattice on a Hilbert space. W hen L is purely atomic, we give a necessary and sufficient condition f or sigma(T) = sigma(L)(T) for every T in algL, where sigma(L)(T) and s igma(T) denote the spectrum of T in algL and B(H) respectively. In add ition, we discuss the properties of the spectra and the invertibility conditions for operators in algL.