Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.23049

Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra

Shih-Yu Chang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Operators with continuous spectra naturally arise in spectral theory, quantum mechanics, automorphic forms, and noncommutative geometry. However, analyzing such operators, particularly in the non-selfadjoint setting, remains challenging due to spectral instability and the lack of an orthonormal basis. This work advances the theory of Multiple Operator Integrals (MOIs) by developing a unified framework for generalized MOIs (GMOIs) associated with general (non-normal, non-selfadjoint) operators possessing continuous spectra. Building on prior work in Generalized Double Operator Integrals (GDOIs) and finite dimensional GMOIs, we extend the theory to include: the formulation of GMOIs in the continuous spectrum setting, their algebraic structure, continuity properties, norm and Lipschitz estimates, and a perturbation formula that generalizes classical results. As a key application, we derive a Krein-type spectral shift formula for GDOIs in the continuous spectrum setting and further extend it to arbitrary-order approximations. These contributions provide a foundation for broader developments in spectral theory, operator algebras, noncommutative geometry, and noncommutative analysis.

Explore related subjects

math.FAmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shih-Yu Chang. 2025-07-30. Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra. https://arxiv.org/abs/2507.23049

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗
arXiv2026

Compactness of Toeplitz Operators on the Bergman Space

Let $φ\in L^\infty(\D)$. We study compactness criteria for \(T_φ\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $φ$, however, its Berezin transform $\tildeφ$ does not readily reveal the intrinsic properties of $φ$. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_φ$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.

math.FA↗
arXiv2026

Order automorphisms of partial isometries in $M_n(\mathbb C)$

We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms considered herein admit no such conventional matrix representations. Instead, they are essentially governed by matrices such that $I-(A+A^*)$ is either positive or negative invertible. The results reveal that the structural features of order automorphisms for partial isometries are substantially more intricate than those of classical subspace automorphisms.

math.FA↗