Operators with closed numerical ranges in nest algebras

Abstract

In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show that Let N be a nest on a Hilbert space H and T is an element of T (N), where T (N) denotes the nest algebra associated with N. Then for given epsilon > 0, there exists a compact operator K with parallel to K parallel to < epsilon such that T + K is an element of T (N) and the numerical range of T + K is closed. As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators.