2024/01/08 by Lin, Huaxin · 1 citation
#46L05 #46L30 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2401.04018
We study the problem when an almost commuting n-tuple self-adjoint operators in an infinite dimensional separable Hilbert space H is close to an n-tuple of commuting self-adjoint operators on H. We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when n≥ 3 is negative even the associated Fredholm index vanishes. In the case that n=2, we show that a pair of almost commuting self-adjoint operators in an infinite dimensional separable Hilbert space is close to a commuting pair of self-adjoint operators if and only if a corresponding Fredholm index vanishes outside of an essential synthetic-spectrum. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements.