strictly singular

Properties.

Strictly singular operators can be viewed as a generalization of compact operators, as every compact operator is strictly singular. These two classes share some important properties. For example, if X is a Banach space and T is a strictly singular operator in B(X) then its spectrum${\displaystyle \sigma (T)}$ satisfies the following properties: (i) the cardinality of ${\displaystyle \sigma (T)}$ is at most countable; (ii) ${\displaystyle 0\in \sigma (T)}$ (except possibly in the trivial case where X is finite-dimensional); (iii) zero is the only possible limit point of ${\displaystyle \sigma (T)}$; and (iv) every nonzero ${\displaystyle \lambda \in \sigma (T)}$ is an eigenvalue. This same "spectral theorem" consisting of (i)-(iv) is satisfied for inessential operators in B(X).