2025/04/24 by Fang, Junsheng, Jiang, Chunlan, Ma, Minghui +3
#46L10 #47C15 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2504.17190
In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in B(H) with respect to the trace-class norm topology? Precisely, for each operator T in B(H) and every ε >0, is there a trace-class operator K such that T+K is irreducible and \Vert K \Vert1 < ε? For p>1, to prove the \Vert ⋅ \Vertp-norm density of irreducible operators in B(H), a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case p=1, where by \Vert ⋅ \Vertp-norm we denote the Schatten p-norm. In the current paper, for a large family of operators in B(H), we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type II1 von Neumann algebras.