Spectral characterization of sums of commutators II

布尔二次型与正切律

The Boolean quadratic forms and tangent law

本文研究布尔交换子与反交换子加权和的极限，证明平移广义正切函数出现在极限定理中，并给出二次型的任意累积量公式，应用于布尔概率论。

Sums of Hermitian Squares as an Approach to the BMV Conjecture

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani (BMV) conjecture that all coefficients of the polynomial p(t)=Tr[(A+tB)^m], where A and B are positive semidefinite matrices of the same size and m an arbitrary integer, are nonnegative. The coefficient of t^k is the trace of S_{m,k}(A,B), which is the sum of all words of length m in the letters A and B in which B appears exactly k times. We consider the case k=4 and show that S_{m,4}(A,B) is a sum of hermitian squares and commutators. In particular, the trace of S_{m,4}(A,B) is nonnegative.

Isomorphisms of spectral lattices

The paper deals with spectral order isomorphisms between certain spectral sublattices of direct sums of AW*-factors. We prove that these maps consist of spectral order isomorphisms between spectral sublattices of individual direct summands. Consequently, we obtain a complete description of spectral order isomorphisms in the case of atomic AW*-algebras. This includes the setting of matrix algebras. Moreover, we also exhibit the general form of spectral order orthoisomorphisms between various spectral sublattices of direct sums of AW*-factors.

Almost everywhere convergence of spectral sums for self-adjoint operators

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} λdE_{ L}(λ)$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(λ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domai…

Localisation of Spectral Sums corresponding to the sub-Laplacian on the Heisenberg Group

In this article we study localisation of spectral sums $\{S_R\}_{R > 0}$ associated to the sub-Laplacian $\mathcal{L}$ on the Heisenberg Group $\mathbb{H}^d$ where $S_R f := \int_0^R dE_{λ}f$, with $\mathcal{L} = \int_0^{\infty} λ\, dE_λ$ being the spectral resolution of $\mathcal{L}.$ We prove that for any compactly supported function $f \in L^2(\mathbb{H}^d)$, and for any $γ< \frac{1}{2}$, $R^γ S_R f \to 0$ as $ R \to \infty$, almost everywhere off $supp (f)$.

Finite-rank approximations of spectral zeta residues

We use the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.

Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral fo…