Cwikel estimates revisited
In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.
math.OA↗
arXiv subjects
Fedor Sukochev
Publications and source records attributed to Fedor Sukochev.
Weak type operator Lipschitz and commutator estimates for commuting tuples
Let $f: \mathbb{R}^d \to\mathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if $\{A_k\}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\in L_1(H),$ then $$\|[f(A_1,\cdots,A_d),B]\|_{1,\infty}\leq c(d)\|\nabla(f)\|_{\infty}\max_{1\leq k\leq d}\|[A_k,B]\|_1,$$ where $c(d)$ is a constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $\{X_k\}_{k=1}^d$ (respectively, $\{Y_k\}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\in L_1(H),$ then $$\|f(X_1,\cdots,X_d)-f(Y_1,\cdots,Y_d)\|_{1,\infty}\leq c(d)\|\nabla(f)\|_{\infty}\max_{1\leq k\leq d}\|X_k-Y_k\|_1.$$
math.OA↗
Johnson-Schechtman inequalities for noncommutative martingales
In this paper we study Johnson-Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space $E(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ without making any assumption on the Boyd indices of $E$. We show that we can obtain Johnson-Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when $E=L_2(0,1)$. As an application, we partly resolve a problem stated by Randrianantoanina and Wu in \cite{NW}. We also show that we can obtain sharp $Φ$-moment analogues for Orlicz functions satisfying $p$-convexity and $q$-concavity for $1 \leq p \leq 2$, $q=2$ and $p=2$, $2< q < \infty$. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder-Gundy inequalities in symmetric spaces and in the $Φ$-moment case.
math.PR↗
Banach limits and traces on $\mathcal L_{1,\infty}$
We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier, the new approach provides the explicit construction of every trace of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on $\mathcal L_{1,\infty}$ and the set of all translation invariant functionals on $l_\infty$. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on $l_\infty$, and definitively classify the measurability of operators in $\mathcal L_{1,\infty}$ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class $\mathcal L_{1,\infty}$) from~\cite{CS}. As an application we extend Connes' classical trace theorem to positive normalised traces.
math.OA↗
Singular traces and perturbation formulae of higher order
Let $H, V$ be self-adjoint operators such that $V$ belongs to the weak trace class ideal. We prove higher order perturbation formula $$τ\big(f(H+V)-\sum_{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{dt^j} f(H+tV)\big|_{t=0}\big)=\int_{\mathbb{R}} f^{(n)}(t)\,dm_n(t),$$ where $τ$ is a trace on the weak trace class ideal and $m_n$ is a finite measure that is not necessarily absolutely continuous. This result extends the first and second order perturbation formulas of Dykema and Shripka, who generalised the Krein and Koplienko trace formulas to the weak trace class ideal. We also establish the perturbation formulae when the perturbation $V$ belongs to the quasi-Banach ideal weak-$L_n$ for any $n \geq 1$.
math.FA↗
Krein's trace theorem revisited
We supply the first proof of Krein's Trace Theorem which does not use complex analysis. Our proof holds for~$σ$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of~$\mathcal{M}$.
math.OA↗
Fubini theorem in noncommutative geometry
We discuss the Fubini formula in Alain Connes' noncommutative geometry. We present a sufficient condition on spectral triples for which a Fubini formula holds true. The condition is natural and related to heat semigroup asymptotics. We provide examples of spectral triples for which the Fubini formula fails.
math.OA↗
Trace Formulas for a Class of non-Fredholm Operators: A Review
We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.
math.AP↗
Johnson-Schechtman and Khinchine inequalities in noncommutative probability theory
We prove disjointification inequalities due to Johnson and Schechtman for noncommutative random variables independent in the sense of Junge and Xu. In the same setting, we also prove noncommutative Khinchine inequalities. These inequalities are proved both for symmetric operator spaces and for modulars.
math.PR↗
$Φ$-moment inequalities for independent and freely independent random variables
This paper is devoted to the study of $Φ$-moments of sums of independent/freely independent random variables. More precisely, let $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let $Φ$ be an Orlicz function with $Δ_2$-condition. We provide an equivalent expression for the quantity $\mathbb{E}(Φ(\sum_{k=1}^n f_k))$ in term of the sum of disjoint copies of the sequence $(f_k)_{k=1}^n.$ We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of $τ(Φ(\sup^+_{1\leq k\leq n}x_k))$ for positive freely independent random variables and also present some new results on free Johnson-Schechtman inequalities in the quasi-Banach symmetric operator space.
math.PR↗
Operator Lipschitz functions on Banach spaces
Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We also study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.
math.FA↗
Double operator integral methods applied to continuity of spectral shift functions
We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $\mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n \to \infty$. Fix $m \in \mathbb{N}$, $m$ odd, $p \in [1,\infty)$, and assume that $T:= \big[( A + iI_{\mathcal{H}})^{-m} - ( B + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, $T_n := \big[( A_n + iI_{\mathcal{H}})^{-m} - ( B_n + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, and $\lim_{n \rightarrow \infty} \|T_n - T\|_{\mathcal{B}_p(\mathcal{H})} =0$. Then for any function $f$ in the class $\mathfrak F_{k}(\mathbb{R}) \supset C_0^{\infty}(\mathbb{R})$ (cf. (1.1)), $$ \lim_{n \rightarrow \infty} \big\| [f(A_n) - f(B_n)] - [f(A)- f(B)]\big\|_{\mathcal{B}_p(\mathcal{H})}=0. $$ Our second result concerns the continuity of spectral shift functions $ξ(\cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $\mathcal{H}$ we denote by $Γ_m(T)$, $m \in \mathbb{N}$ odd, the set of all self-adjoint operators $S$ in $\mathcal{H}$ satisfying $\big[(S - z I_{\mathcal{H}})^{-m} - (T - z I_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_1(\mathcal{H})$, $z \in \mathbb{C}\backslash \mathbb{R}$. Employing a suitable topology on $Γ_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1\in Γ_m(B_0)$ and let $\{B_τ\}_{τ\in [0,1]}\subset Γ_m(B_0)$ denote a path from $B_0$ to $B_1$ in $Γ_m(B_0)$ depending continuously on $τ\in [0,1]$ with respect to the topology on $Γ_m(B_0)$. If $f \in L^{\infty}(\mathbb{R})$, then $$ \lim_{τ\to 0^+} \|ξ(\, \cdot \, ; B_τ, A_0) f - ξ(\, \cdot \, ; B_0, A_0) f\|_{L^1(\mathbb{R}; (|ν|^{m+1} + 1)^{-1}dν)} = 0. $$
math.SP↗
An upper triangular decomposition theorem for some unbounded operators affiliated to II_1-factors
Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the Brown measure of the operator.
math.OA↗
Algebras of log-integrable functions and operators
We show that certain spaces of log-integrable functions and operators are complete topological *-algebras with respect to a natural metric space structure. We explore connections with the Nevanlinna class of holomorphic functions.
math.OA↗
On reduction theory and Brown measure for closed unbounded operators
The theory of direct integral decompositions of both bounded and unbounded operators is further developed; in particular, results about spectral projections, functional calculus and affiliation to von Neumann algebras are proved. For operators belonging to or affiliated to a tracial von Neumann algebra that is a direct integral von Neumann algebra, the Brown measure is shown to be given by the corresponding integral of Brown measures.
math.OA↗
On Index Theory for Non-Fredholm Operators: A $(1+1)$-Dimensional Example
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(\mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = \int^{\oplus}_{\mathbb{R}} dt \, A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(\mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + θ(t) ϕ(\cdot)$, $t \in \mathbb{R}$. Here $ϕ: \mathbb{R} \to \mathbb{R}$ has to be integrable on $\mathbb{R}$ and $θ: \mathbb{R} \to \mathbb{R}$ tends to zero as $t \to - \infty$ and to $1$ as $t \to + \infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + ϕ(\cdot)$ as $t \to \mp \infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = \lim_{λ\to 0} (- λ) {\rm tr}_{L^2(\mathbb{R}^2; dtdx)}((H_1 - λI)^{-1} - (H_2 - λI)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = ξ(0_+; H_2, H_1) = ξ(0; A_+, A_-) = 1/(2 π) \int_{\mathbb{R}} dx \, ϕ(x). $$ Here $ξ(\, \cdot \, ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $ξ(λ; H_2, H_1) = 0$, $λ< 0$.
math-ph↗
On the Index of a Non-Fredholm Model Operator
Let $\{A(t)\}_{t \in \mathbb{R}}$ be a path of self-adjoint Fredholm operators in a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$ as $t \to \pm \infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A = \int_{\mathbb{R}}^{\oplus} dt \, A(t)$, and its relation to spectral flow along this path, has a long history. While most of the latter focuses on the case where $A(t)$ all have purely discrete spectrum, we now particularly study situations permitting essential spectra. Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we consider spectral shift functions $ξ(\, \cdot \,; A_+, A_-)$ and $ξ(\, \cdot \, ; H_2, H_1)$ associated with the pairs $(A_+, A_-)$ and $(H_2,H_1)$. Assuming $A_+$ to be a relatively trace class perturbation of $A_-$ and $A_{\pm}$ to be Fredholm, the value $ξ(0; A_-, A_+)$ was shown in [14] to represent the spectral flow along the path $\{A(t)\}_{t\in \mathbb{R}}$ while that of $ξ(0_+; H_1,H_2)$ yields the Fredholm index of $D_A$. The fact, proved in [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. When the path $\{A(t)\}_{t \in \mathbb{R}}$ consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in (1+1)-dimensions) is a relatively Hilbert-Schmidt perturbation. This is not just an incremental improvement. In fact, the approximation method we employ here to make this extension is of interest in any dimension. Moreover we consider $A_\pm$ which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions for the pairs $(A_+, A_-)$ and $(H_2,H_1)$ found in all of the previous papers [9], [14], and [22] can be proved in the non-Fredholm case.
math.SP↗
Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case
We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.
math.FA↗