On a boundedness condition for operators with a singleton spectrum

Abstract

For a bounded invertible linear operator A let B A {\mathcal {B}_A} consist of those operators X for which sup { ‖ A n X A − n ‖ : n ⩾ 0 } > ∞ \sup \{ \left \| {{A^n}X{A^{ - n}}} \right \|:n \geqslant 0\} > \infty . It is shown that B A {\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 B A ∩ B A − 1 {\mathcal {B}_A} \cap {\mathcal {B}_{{A^{ - 1}}}} is the commutant of A.