arXiv subjects

Fedor Sukochev

Publications and source records attributed to Fedor Sukochev.

At least 109 records · Page 6Linked to original sources

arXiv2015

Catalysis in the trace class and weak trace class ideals

Given operators $A,B$ in some ideal $\mathcal{I}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on a separable Hilbert space $H$, can we give conditions guaranteeing the existence of a trace-class operator $C$ such that $B \otimes C$ is submajorized (in the sense of Hardy--Littlewood) by $A \otimes C$ ? In the case when $\mathcal{I} = \mathcal{L}_1$, a necessary and almost sufficient condition is that the inequalities ${\rm Tr} (B^p) \leq {\rm Tr} (A^p)$ hold for every $p \in [1,\infty]$. We show that the analogous statement fails for $\mathcal{I} = \mathcal{L}_{1,\infty}$ by connecting it with the study of Dixmier traces.

math.FA↗

arXiv2015

Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a Lipschitz function. If $A,B\in\mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M}),$ then $$\|[f(A),B]\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|[A,B]\|_1,$$ where $c_{abs}$ is an absolute constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $X,Y\in\mathcal{M}$ are self-adjoint operators such that $X-Y\in L_1(\mathcal{M}),$ then $$\|f(X)-f(Y)\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|X-Y\|_1.$$ This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.

math.OA↗

arXiv2015

The Spectral shift function and the Witten index

We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53] and develop its extension to general $σ$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of $\mathcal{M}$. We also discuss the connection between the theory of the spectral shift function and index theory for certain model operators. We start by introducing various definitions of the Witten index, (an extension of the notion of Fredholm index to non-Fredholm operators). Then we study the model operator $D_{A^{}} = (d/dt) + A$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$. The setup permits the operator family $A(t)$ on $\mathcal{H}$ to be an unbounded relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on the asymptotes $A_{\pm}$. When $A_{\pm}$ are boundedly invertible, it is shown that $D_{A^{}}$ is Fredholm and its index can be computed as $ξ(0; A_+, A_-)$. When $0\inσ(A_+)$ (or $0\inσ(A_-)$), the operator $D_{A^{}}$ ceases to be Fredholm. However, if $0$ is a right and a left Lebesgue point of $ξ(\,\cdot\,\, ; A_+, A_-)$, the resolvent regularized Witten index $W_r(D_{A^{}})$ is given by $W_r(D_{A^{}}) = ξ(0_+; |D_{A^{*}}|^2, |D_{A^{}}|^2) = [ξ(0_+; A_+,A_-) + ξ(0_-; A_+, A_-)]/2$. We also study a special example, when the perturbation of the unbounded self-adjoint operator $A_-$ is not assumed to be relatively trace class.

math.SP↗

arXiv2015

Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$

math.FA↗

arXiv2015

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A = (d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$, via the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ of asymptotic operators $A_{\pm}=A(\pm\infty)$ on the separable complex Hilbert space $\mathcal{H}$ in the case when $A(t)$ is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator $A_-$. We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ for the pair $(A_+, A_-)$, and the corresponding spectral shift function $ξ(\, \cdot \,;H_2,H_1)$ for the pair of operators $(H_2,H_1)=(D_A {D_A}^*, {D_A}^* D_A)$ in this relative trace class context. This formula is then used to identify the Fredholm index of $D_A$ with $ξ(0;A_+,A_-)$. In addition, we prove that ${\rm ind}(D_A)$ coincides with the spectral flow SpFlow$(\{A(t)\}_{t=-\infty}^\infty)$ of the family $\{A(t)\}_{t\in\mathbb{R}}$ and also relate it to the (Fredholm) perturbation determinant for the pair $(A_+, A_-)$. We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.

math.SP↗

arXiv2014

On the Witten index in terms of spectral shift functions

We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$ (with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$ the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits $A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on $A_{\pm}$. We introduce resolvent and semigroup regularized Witten indices of $\mathbf{D}_{\mathbf{A}}$, denoted by $W_r$ and $W_s$, and prove that these regularized indices coincide with the Fredholm index of $\mathbf{D}_{\mathbf{A}}$ whenever the latter is Fredholm. In situations where $\mathbf{D}_{\mathbf{A}}$ ceases to be a Fredholm operator in $L^2(\mathbb{R};\mathcal{H})$ we compute its resolvent (resp., semigroup) regularized Witten index in terms of the spectral shift function $ξ(\,\cdot\,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ as follows: Assuming $0$ to be a right and a left Lebesgue point of $ξ(\,\cdot\,\, ; A_+, A_-)$, denoted by $ξ_L(0_+; A_+,A_-)$ and $ξ_L(0_-; A_+, A_-)$, we prove that $0$ is also a right Lebesgue point of $ξ(\,\cdot\,\, ; \mathbf{H_2}, \mathbf{H_1})$, denoted by $ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1})$, and that \begin{align*} W_r(\mathbf{D}_{\mathbf{A}}) &= W_s(\mathbf{D}_{\mathbf{A}}) \\ & = ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1}) \\ & = [ξ_L(0_+; A_+,A_-) + ξ_L(0_-; A_+, A_-)]/2, \end{align*} the principal result of this paper. In the special case where $\dim(\mathcal{H}) < \infty$, we prove that the Witten indices of $\mathbf{D}_{\mathbf{A}}$ are either integer, or half-integer-valued.

math.SP↗

arXiv2014

The best constants for operator Lipschitz functions on Schatten classes

Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in B(H) is an operator such that the commutator [A, x] is contained in the Schatten class S_p. It is proved by the last two authors, that then also [f(A), x] is contained in S_p and there exists a constant C_p independent of x and f such that || [f(A), x] ||_p <= C_p || [A,x] ||_p. The main result of this paper is to give a sharp estimate for C_p in terms of p. Namely, we show that C_p ~ p^2/(p-1). In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of n self-adjoint operators, for which we prove the analogous result. The case described here in the abstract follows as a special case.

math.FA↗

arXiv2014

Some Operator Bounds Employing Complex Interpolation Revisited

We revisit and extend known bounds on operator-valued functions of the type $$ T_1^{-z} S T_2^{-1+z}, \quad z \in \ol Σ= \{z\in\bbC\,|\, \Re(z) \in [0,1]\}, $$ under various hypotheses on the linear operators $S$ and $T_j$, $j=1,2$. We particularly single out the case of self-adjoint and sectorial operators $T_j$ in some separable complex Hilbert space $\cH_j$, $j=1,2$, and suppose that $S$ (resp., $S^*$) is a densely defined closed operator mapping $\dom(S) \subseteq \cH_1$ into $\cH_2$ (resp., $\dom(S^*) \subseteq \cH_2$ into $\cH_1$), relatively bounded with respect to $T_1$ (resp., $T_2^*$). Using complex interpolation methods, a generalized polar decomposition for $S$, and Heinz's inequality, the bounds we establish lead to inequalities of the following type, \begin{align*} & \big\|\ol{T_2^{-x}ST_1^{-1+x}}\big\|_{\cB(\cH_1,\cH_2)} \leq N_1 N_2 e^{(θ_1 + θ_2) [x(1-x)]^{1/2}} \\ & \quad \times \big\|ST_1^{-1}\big\|_{\cB(\cH_1,\cH_2)}^{1-x} \, \big\|S^*(T_2^*)^{-1}\big\|_{\cB(\cH_2,\cH_1)}^{x}, \quad x \in [0,1], \end{align*} assuming that $T_j$ have bounded imaginary powers, that is, for some $N_j\ge 1$ and $θ_j \ge 0,$ $$ \big\|T_j^{is}\big\|_{\cB(\cH)} \leq N_j e^{θ_j |s|}, \quad s \in \bbR, \; j=1,2. $$ We also derive analogous bounds with $\cB(\cH_1,\cH_2)$ replaced by trace ideals, $\cB_p(\cH_1, \cH_2)$, $p \in [1,\infty)$. The methods employed are elementary, predominantly relying on Hadamard's three-lines theorem and Heinz's inequality.

math.FA↗

arXiv2014

A Jost-Pais-type reduction of Fredholm determinants and some applications

We study the analog of semi-separable integral kernels in $\cH$ of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \cB_2(\cH_j,\cH)$ and $G_j(x) \in \cB_2(\cH,\cH_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with $\cH$ and $\cH_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a trace class operator $\bsK$ in $L^2((a,b);\cH)$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant ${\det}_{L^2((a,b);\cH)}(\bsI - α\bsK)$, $α\in \bbC$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\cH$ (and $\cH_1 \oplus \cH_2$). Explicit applications of this reduction theory are made to Schrödinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.

math.FA↗

arXiv2014

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to Hörmander-Weyl pseudo-differential calculus. In that context, we prove that the Dixmier trace of a pseudo-differential operator coincide with the `Dixmier integral' of its symbol.

math.OA↗

arXiv2013

Holomorphic functional calculus on upper triangular forms in finite von Neumann algebras

The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup--Schultz hyperinvariant projections, behave well with respect to holomorphic functional calculus.

math.OA↗

arXiv2013

Frechet differentiability of Sp norms

One of the long standing questions in the theory of Schatten-von Neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts. We answer this question in the affirmative. A key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of L.S. Koplienko's conjecture concerning existence of higher order spectral shift functions.

math.OA↗

arXiv2013

On a conjecture of A. Bikchentaev

In \cite{bik1}, A. M. Bikchentaev conjectured that for positive $τ-$measurable operators $a$ and $b$ affiliated with an arbitrary semifinite von Neumann algebra $\mathcal M$, the operator $b^{1/2}ab^{1/2}$ is submajorized by the operator $ab$ in the sense of Hardy-Littlewood. We prove this conjecture in full generality and present a number of applications to fully symmetric operator ideals, Golden-Thompson inequality and (singular) traces.

math.OA↗

arXiv2012

Spectral shift function of higher order

This paper resolves affirmatively Koplienko's conjecture of 1984 on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions $η_n$. We show the higher order spectral shift function is a $L^1$-function and prove an estimate on its $L^1$-norm. Existence and summability of $η_1$ and $η_2$ were established by Krein in 1953 and Koplienko in 1984, respectively, whereas for $n > 2$ the problem was unresolved. Our method is derived from [arXiv:0904.4095]; it also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.

math.FA↗

arXiv2012

Higher order spectral shift for contractions

We derive strong estimates for Schatten norms of operator derivatives along paths of contractions and apply them to prove existence of higher order spectral shift functions for pairs of contractions.

math.FA↗

arXiv2012

Traces of compact operators and the noncommutative residue

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order $-d$ and generalise A. Connes' trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order $-d$ do not have a `unique' trace; pseudo-differential operators can be non-measurable in Connes' sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry à la Connes, including the definitive statement of Connes' original theorem.

math.FA↗

arXiv2012

On the distinction between the classes of Dixmier and Connes-Dixmier traces

In the present paper we prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal $\mathcal M_{1,\infty}$. We construct a Marcinkiewicz space $\mathcal M_ψ$ and a positive operator $T\in \mathcal M_ψ$ which is Connes-Dixmier measurable but which is not Dixmier measurable.

math.OA↗

arXiv2012

Dixmier traces generated by exponentiation invariant generalised limits

We define a new class of singular positive traces on the ideal $\mathcal M_{1,\infty}$ of $B(H)$ generated by exponentiation invariant generalized limits. We prove that this new class is strictly contained in the class of all Dixmier traces. We also prove a Lidskii-type formula for this class of traces.

math.OA↗