On the Commutant of Invertible Positive Operators
An important result due to Radjabalipour asserts that if ${A\in\BB(\H)}$ is an invertible positive operator and ${T\in\BB(\H)}$ is arbitrary, then the operator sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded if and only if ${TA=AT}.$ Using the concept of spectral gaps, we show that if $A$ possesses a spectral gap and the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded, then $T$ must be block diagonal with respect to a certain orthogonal decomposition of $\H$. We also provide an alternative proof of Radjabalipour's theorem. We also extend the theorem to some rich classes of operators, including invertible normal operators and invertible weighted sums of projections. Moreover, we provide simple proofs for the fact that a positive definite matrix ${A\in\MM_N}$ is a positive multiple of the identity if and only if the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded for every~${T\in\MM_N}.$