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Galina Levitina

Publications and source records attributed to Galina Levitina.

14 recordsLinked to original sources
arXiv2024

Symmetric functionals on simply generated symmetric spaces

In the present paper we suggest a construction of symmetric functionals on a large class of symmetric spaces over a semifinite von Neumann algebra. This approach establishes a bijection between the symmetric functionals on symmetric spaces and shift-invariant functionals on the space of bounded sequences. It allows to obtain a bijection between the classes of all continuous symmetric functionals on different symmetric spaces. Notably, we show that this mapping is not bijective on the class of all Dixmier traces. As an application of our results we prove an extension of the Connes trace formula for a wide class of operators and symmetric functionals.

math.OA↗
arXiv2022

The Witten index and the spectral shift function

In \cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon \cite{RS95}. In \cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper \cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0\leq p<\infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by \cite{BCPRSW, CGK16}. We illustrate our results using Dirac type operators on $L^2(\bbR^d)$ for arbitrary $d\in\bbN$. In this setting our main result substantially extends \cite[Theorem 3.5]{CGGLPSZ16}, where the case $d=1$ was treated.

math.FA↗
arXiv2021

The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

We derive a limiting absorption principle on any compact interval in $\mathbb{R} \backslash \{0\}$ for the free massless Dirac operator, $H_0 = α\cdot (-i \nabla)$ in $[L^2(\mathbb{R}^n)]^N$, $n \geq 2$, $N=2^{\lfloor(n+1)/2\rfloor}$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H = H_0 +V$, where $V$ decays like $O(|x|^{-1 - \varepsilon})$. Expressing the spectral shift function $ξ(\,\cdot\,; H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $ξ(\,\cdot\,;H,H_0) \in C((-\infty,0) \cup (0,\infty))$, and that the left and right limits at zero, $ξ(0_{\pm}; H,H_0)$, exist. Introducing the non-Fredholm operator $\boldsymbol{D}_{\boldsymbol{A}} = \frac{d}{dt} + \boldsymbol{A}$ in $L^2\big(\mathbb{R};[L^2(\mathbb{R}^n)]^N\big)$, where $\boldsymbol{A} = \boldsymbol{A_-} + \boldsymbol{B}$, $\boldsymbol{A_-}$, and $\boldsymbol{B}$ are generated in terms of $H, H_0$ and $V$, via $A(t) = A_- + B(t)$, $A_- = H_0$, $B(t)=b(t) V$, $t \in \mathbb{R}$, assuming $b$ is smooth, $b(-\infty) = 0$, $b(+\infty) = 1$, and introducing $\boldsymbol{H_1} = \boldsymbol{D}_{\boldsymbol{A}}^{*} \boldsymbol{D}_{\boldsymbol{A}}$, $\boldsymbol{H_2} = \boldsymbol{D}_{\boldsymbol{A}} \boldsymbol{D}_{\boldsymbol{A}}^{*}$, one of the principal results in this manuscript expresses the $k$th resolvent regularized Witten index $W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}})$ ($k \in \mathbb{N}$, $k \geq \lceil n/2 \rceil$) in terms of spectral shift functions as \[ W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}) = ξ(0_+; \boldsymbol{H_2}, \boldsymbol{H_1}) = [ξ(0_+;H,H_0) + ξ(0_-;H,H_0)]/2. \] Here $L^2(\mathbb{R};\mathcal{H}) = \int_{\mathbb{R}}^{\oplus} dt \, \mathcal{H}$ and $\boldsymbol{T} = \int_{\mathbb{R}}^{\oplus} dt \, T(t)$ abbreviate direct integrals.

math.SP↗
arXiv2020

Pietsch correspondence for symmetric functionals on Calkin operator spaces associated with semifinite von Neumann algebras

In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We prove that a shift-monotone space $E(\Z)$ of sequences indexed by $\Z$ defines a Calkin space $E(\cM,τ)$ of $τ$-measurable operators affiliated with a semifinite von Neumann algebra $\cM$ equipped with a faithful normal semifinite trace $τ$. Furthermore, we show that shift-invariant functionals on $E(\Z)$ generate symmetric functionals on $E(\cM,τ)$. In the special case, when the algebra $\cM$ is atomless or atomic with atoms of equal trace, the converse also holds and we have a bijective correspondence between all shift-monotone spaces $E(\Z)$ and Calkin spaces $E(\cM,τ)$ as well as a bijective correspondence between shift-invariant functionals on $E(\Z)$ and symmetric functionals on $E(\cM,τ)$. The bijective correspondence $E(\Z)\leftrightarrows E(\cM,τ)$ extends to a correspondence between complete symmetrically $Δ$-normed spaces $E(\cM,τ)$ and complete $Δ$-normed shift-monotone spaces $E(\Z)$.

math.OA↗
arXiv2019

Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.

math.OA↗
arXiv2017

On the Global Limiting Absorption Principle for Massless Dirac Operators

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = α\cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.

math.SP↗
arXiv2017

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↗
arXiv2016

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↗
arXiv2016

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↗
arXiv2015

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↗
arXiv2015

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↗
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↗