We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asympt
otics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates
for integral operators with kernel in $B^{frac{1}{p}-frac{1}{2}}_{p,p}(mathbb R,L_2(mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{frac{1}{p}-frac{1}{2}}_{frac{2p}{2-p},p}(mathbb R,L_infty(mathbb R))$, which extends Birman and Solomyaks result to symbols without compact domain.
We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distribution
A version of Connes Integration Formula which provides concrete asymptotics of the eigenvalues is given. This radically extending the class of quantum-integrable functions on compact Riemannian manifolds.
Let $mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $tau$. L
et $E(mathcal{M},tau) $ be a symmetric operator space affiliated with $ mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $left|cdotright|_2$ on $L_2(mathcal{M},tau)$. We obtain general description of all bounded hermitian operators on $E(mathcal{M},tau)$. This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.
We study the class of functions $f$ on $mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $mathcal{L}_p$ for $0 < p leq 1$. The corresponding problem with $pgeq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is
by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})$ obey the estimate $$ |f(A)-f(B)|_{p} leq C_{p}(|f|_{L_{infty}(mathbb{R})}+|f|_{dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})})|A-B|_{p} $$ for all bounded self-adjoint operators $A$ and $B$ with $A-Bin mathcal{L}_p$. In the case $p=1$, our methods recover and provide a new perspective on a result of Peller that $f in dot{B}^1_{infty,1}$ is sufficient for a function to be Lipschitz in $mathcal{L}_1$. We also provide related Holder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on $mathbb{R}$ are not Lipschitz in $mathcal{L}_p$ for any $0 < p < 1$. This gives counterexamples to a 1991 conjecture of Peller that $f in dot{B}^{1/p}_{infty,p}(mathbb{R})$ is sufficient for $f$ to be Lipschitz in $mathcal{L}_p$.
We present a new approach to Lorentz-Shimogaki and Arazy-Cwikel Theorems which covers all range of $p,qin (0,infty]$ for function spaces and sequence spaces. As a byproduct, we solve a conjecture of Levitina and the last two authors.
This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and [Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of $L_{p,q}$, for $1< p<infty$, $1le q<infty$, $p e q $, on resonant measure spaces, has been obtained.
The aim of this paper is to give a complete isomorphic classification of $L_{p,q}$-spaces on general $sigma$-finite measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and $L_{p,q}(0,infty)$ are identified and studied.
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrodinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of s
tates. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.
