Abstract
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.
Highlights
Introduction and PreliminariesLie algebra is a standard language for continuous symmetry, while operator algebra is a foundational language for quantum physics, and they usually interact with each other
In 1960, generalizing von Neumann’s bicommutant theorem in the matrix case, M
We mainly prove that for any bounded spectral operator A = S + N on a complex Hilbert space H, if the radical part N is locally nilpotent, i.e., if
Summary
Lie algebra is a standard language for continuous symmetry, while operator algebra is a foundational language for quantum physics, and they usually interact with each other. If we prove this lemma for any normal operator A, the general case readily follows, observing adsP−1 AP(X) = P−1adsA(PXP−1)P. In [1], the assumption is weakened, i.e., claiming that it only needs to be essentially nilpotent, and some necessary and sufficient conditions for an essentially nilpotent Lie algebra of quasi-nilpotent operators to generate the closed algebra of quasi-nilpotent operators are given From these results, we see that the irreducible module H over the nilpotent Lie algebra N is 1-dimensional, which is closely linked with Lie’s Theorem in Lie theory If A is a compact self-adjoint operator on H, A has the canonical spectral decomposition ( known as diagonalization) A = ∑i∞=1 λiPi (see Theorem 5.1, Chapter II in [18]).
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