On a Boundedness Condition for Operators with a Singleton Spectrum

Abstract

For a bounded invertible linear operator A let ${\mathcal {B}_A}$ consist of those operators X for which $\sup \{ \left \| {{A^n}X{A^{ - n}}} \right \|:n \geqslant 0\} > \infty$. It is shown that ${\mathcal {B}_A}$ contains the ideal of compact operators if and only if A is similar to a scalar multiple of a unitary operator. Also, if A is invertible and either has a one-point spectrum or is positive definite then ${\mathcal {B}_A} \cap {\mathcal {B}_{{A^{ - 1}}}}$ is the commutant of A.