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In this short article, we mainly prove that, for any spectral operator $A$ oftype $m$ on a complex Hilbert space, if a bounded operator $B$ lies in thecollection of bounded linear operators that are in the $k$-centralizer of everybounded linear operator in the $l$-centralizer of $A$, where $k\leqslant l$ istwo 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 theoremproved by M.\ F.\ Smiley. For this aim, Smiley-type operators are defined andstudied.

A Smiley-type theorem for spectral operators of finite type