Density of irreducible operators in the trace-class norm
Junsheng Fang, Chunlan Jiang, Minghui Ma, Junhao Shen, Rui Shi, Tianze Wang
TL;DR
The paper resolves, for a large family of operators, the long-standing problem of approximating any $T\in\mathcal{B}(\mathcal{H})$ by trace-class perturbations $K$ so that $T+K$ is irreducible, by connecting the density question to the structure of type $\mathrm{II}_1$ von Neumann algebras. It develops a novel blend of perturbation theory in the trace-class norm with single-generator techniques and von Neumann-algebra methods, including generating-vectors in properly infinite algebras, atomic supports, and Cartan subalgebras. A central result shows that if $W^*(T)$ is a type $\mathrm{II}_1$ factor, one can achieve $T+K$ as a direct sum of irreducibles for arbitrarily small $\|K\|_1$, and the authors outline equivalences tying density to a conjecture about II$_1$ generators. The work thus links the density problem to deep operator-algebraic properties, provides concrete perturbation mechanisms, and maps out several operator-classes that lie in the trace-class closure of irreducibles, with potential implications for Halmos-type density questions and the study of irreducible operators through von Neumann algebraic structure.
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.