Density of irreducible operators in the trace-class norm

Abstract

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 $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$? For $p>1$, to prove the $\Vert \cdot \Vert_p$-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 $\Vert \cdot \Vert_p$-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 $\mathrm{II}_1$ von Neumann algebras.