On a generalization of Smiley's theorem

Xiao Chen, Jian-Jian Jiang, Xiaolin Li

Published 2021-07-24Version 1

In this short article, we mainly prove that, for any bounded operator $A$ of the form $S+N$ on a complex Hilbert space, where $S$ is a normal operator and $N$ is a nilpotent operator commuting with $S$, if a bounded operator $B$ lies in the collection of bounded linear operators that are in the $k$-centralizer of every bounded linear operator in the $l$-centralizer of $A$, where $k\leqslant l$ is two arbitrary positive integers, then $B$ must belong to the operator norm closure of the algebra generated by $A$ and identity operator. This result generalizes a matrix commutator theorem proved by M.\ F.\ Smiley. For this aim, Smiley-type operators are defined and studied. Furthermore, we also give a generalization of Fuglede's theorem as a byproduct.