Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1710.10680

Weyl's Theorem for pairs of commuting hyponormal operators

Sameer Chavan·Raul E. Curto
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbolλ) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbolλ})^*), $$ for every $\boldsymbolλ$ in the Taylor spectrum $σ(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $ω(\mathbf{T})$, satisfies the identity $$ ω(\mathbf{T})=σ(\mathbf{T}) \setminus π_{00}(\mathbf{T}), $$ where $π_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity. Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sameer Chavan, Raul E. Curto. 2017-10-29. Weyl's Theorem for pairs of commuting hyponormal operators. https://doi.org/10.1090/proc%2F13479

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

KEEP EXPLORING

Related papers

arXiv2026

Conditional expectation operators on $C(X)$

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.

math.FA↗
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

Logarithmic oscillatory multipliers and log-subdyadic square functions

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ ρ(R)=\frac{R}{(\log R)^{γ-1}}, \qquad γ>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $ρ(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \[ L(ξ)=\frac12\log(e^2+|ξ|^2), \qquad m_{γ,β}(ξ)=L(ξ)^{-β}e^{iL(ξ)^γ}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $β\geq0$. For $1 d(γ-1)\left|\frac12-\frac1p\right|. \] At the critical equality we obtain the corresponding Lorentz endpoint estimates.

math.FA↗