  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.
