The main finding is a sharp divide between finite-rank and compact perturbations, mathematical changes applied to an operator. Finite-rank perturbations preserve membership in β0(H), the class whose modulus belongs to β(H). The guarantee does not extend to arbitrary compact perturbations: the paper gives a trace-class example in which the resulting operator falls outside β0(H). The largest subset of β0(H) that remains stable under every compact perturbation is K(H), the compact-operator class.
The spectral test behind the classes
The work is entirely theoretical. It studies operators on an infinite-dimensional separable complex Hilbert space H and the bounded-operator algebra B(H), with no empirical sample. In the paper’s notation, β(H) is the class in which an operator’s modulus attains its norm on every nonzero reducing subspace. β0(H) is defined through that class: its modulus must itself belong to β(H).
The paper’s corrected structural test for β0(H) is exact. An operator T belongs to β0(H) precisely when its modulus |T| has pure point spectrum and its point spectrum, the set of eigenvalues, satisfies a reverse well-ordering condition. This is a necessary-and-sufficient description of membership, not a sufficient pattern that happens to work for some examples.
Where stability breaks
That test drives the finite-rank result. When T is in β0(H) and K is finite-rank, T+K remains in β0(H), and hence in β(H). The theorem therefore gives a firm stability result for finite-rank perturbations, while it does not provide the same blanket guarantee for the broader compact case.
The broader case is where the boundary appears. The supplied counterexample uses a trace-class perturbation yet produces an operator outside β0(H), so compactness alone is not enough for automatic stability. Instead, the paper gives a compact-perturbation test: T+K belongs to β0(H) exactly when the total perturbation operator L has pure point spectrum and a reverse well-ordered point spectrum.
Under an additional condition, that test becomes shorter. If D is compact, T+K belongs to β0(H) exactly when the point-spectrum values of L below zero are finite in number. The examples are presented as showing that both the finiteness of the negative eigenvalues and the compactness of D are sharp requirements, including for perturbations in every Schatten p-class.
Together, these results identify the maximal stable core. K(H) is the largest subset of β0(H) that survives every compact perturbation. That does not replace the case-by-case criterion for a particular compact K, but it sets the largest uniform guarantee described by the analysis.
A wide approximation result
A related result concerns how the modulus can be used to classify β(H). The paper establishes the implication from |T| in β(H) to T in β(H), while the converse fails in general. For a general T, membership in β(H) is itself equivalent to |T| attaining its norm on every reducing subspace of T. The implication is therefore one-way, even though the modulus remains central to both descriptions.
The perturbation limits do not prevent a broad approximation result. β(H) is dense in B(H) under the operator norm, so bounded operators can be approximated arbitrarily closely in that norm by members of β(H). The proof first approximates the modulus on the orthogonal complement of the kernel of T with a finite-spectrum self-adjoint operator, then uses the polar decomposition to build the approximating operator.
A result in an abstract setting
The simpler negative-spectrum criterion depends on D being compact, and the full compact criterion depends on the spectral structure of L. The work is a preprint; its front matter identifies arXiv version 2 dated 30 September 2026 and leaves the received, revised and accepted fields as placeholders.
Paper data and sources
Original title: The Stability and Density of Norm Attaining Operators on Reducing Subspaces
Authors: Zinan Su, Yuanhang Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text