A Smiley-type theorem for spectral operators of finite type

TL;DR

This paper generalizes a matrix commutator theorem to spectral operators of finite type on complex Hilbert spaces, establishing conditions under which certain operators belong to the von Neumann algebra generated by a spectral operator.

Contribution

It introduces Smiley-type operators and proves a generalized commutator theorem for spectral operators of finite type, extending prior matrix results.

Findings

Operators in specific centralizers belong to the von Neumann algebra generated by the spectral operator.

The result applies to spectral operators of finite type on complex Hilbert spaces.

Generalizes Smiley's matrix commutator theorem to a broader operator setting.

Abstract

In this short article, we mainly prove that, for any spectral operator $A$ of type $m$ on a complex Hilbert space, 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 satisfying $l\geqslant k$ as well as $l\geqslant 2m+1$, then $B$ must belong to the von Neumann 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.