The radius of the essential spectrum (Q5903615)

An operator measure s is introduced axiomatically as a map from L(X) (bounded linear operators on the Banach space X into the nonnegative real numbers) which vanishes on all finite rank operators and only on compact operators, is dominated by the norm function and is subadditive and submultiplicative. The author considers the quotient algebra L(X)/I where I is the closed ideal of all operators T with \(s(T)=0\), and defines the I-essential spectrum \(\sigma_ I(T)=\{\lambda \in {\mathbb{C}}:\) the equivalence class of \(\lambda\)-T is not invertible in L(X)/I\(\}\). It turns out that \(\sigma_ 1(T)\) is equal to the essential spectrum of T and the radius \(r_ e(T)\) of the essential spectrum of T can be recovered as \(\lim_{n}(s(T^ n))^{1/n}\). Examples of such operator measures s are also given.