Smiley Theorem for Spectral Operators Whose Radical Part Is Locally Nilpotent

Generalizing bicommutant theorem to the higher-order commutator case is very useful for representation theory of Lie algebras, which plays an important role in symmetry analysis. In this paper, we mainly prove that for any spectral operator A on a complex Hilbert space whose radical part is locally nilpotent, if a bounded operator B lies in the k-centralizer of every bounded linear operator in the l-centralizer of A, where k and l are two arbitrary positive integers satisfying l⩾k, then B must belong to the von Neumann algebra generated by A and the identity operator. This result generalizes a matrix commutator theorem proved by M. F. Smiley. To this aim, Smiley operators are defined and an example of a non-spectral Smiley operator is given by the unilateral shift, indicating that Smiley-type theorems might also hold for general spectral operators.