We study the spectral functional tr f(D+A) for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives.
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 f…
We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being dissipative or bounded, and show that these measures can have singular components in the case of Dixmier traces. We also establish a linearization formula for a Dixmier trace applied to perturbed operator functions, a result that does not typically hold for normal traces.
We discuss applications of the M. G. Kre\uın theory of the spectral shift function to the multi-dimensional Schrödinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace identities for the Schrödinger operator are derived.
We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained. This formula includes the Livshits--Krein trace formula in the case $m=1$ and the Koplienko trace formula in the case $m=2$. We establish most general trace formulae in the case of perturbation of Schatten--von Neumann class $\bS_m$. We also improve the trace formula obtained in \cite{PSS} for operator Taylor polynomials and prove it for arbitrary functions …
We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$ generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl operators and the lifts of the unitary operators on $\mathbb C^n$ to metaplectic operators. With the help of an auxiliary operator in the Shubin calculus, we find trace expansions for these operators in the spirit of Grubb and Seeley. Moreover, we can define a noncommutative residue generalizing that for the Shubin pseudodifferential operators and obtain a class of localized equivariant traces on the algebra.
Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace inequalities for matrices are also derived. Examples for the operator exponential and other similar functions are presented as well.
The paper deals with trace operators of Wiener amalgam spaces using frequency-uniform decomposition operators and maximal inequalities, obtaining sharp results. Additionally, we provide the embeddings between standard and anisotropic Wiener amalgam spaces.
We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kre\uın type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.
We characterize the trace class membership of Toeplitz operators with distributional symbols acting on the Bergman space on the unit disk. The Berezin transform of distributions, introduced in the paper, yields a formula for the trace. Several instructive examples are also given.
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its dual group $\widehat{G}$. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class $j$-Weyl transform, $j \in \mathbb{Z}^*$ with symbols in $L^{2}(G\times \widehat{G}).$ We go on to present a characterization of t…
This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.
This paper is concerned with a boundedness of trace and extension operators for Besov spaces and Triebel-Lizorkin spaces on upper half space with variable exponents. To define trace and extension operators, we introduce a quarkonial decomposition for Besov spaces and Triebel-Lizorkin spaces with variable exponents on ${\mathbb R}^n$. Furthermore, we study trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents on upper half spaces ${\mathbb R}^n_+$.
The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz functions on the unit circle.
The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions coincides with the class of operator Lipschitz functions.
By the use of the celebrated Kato's inequality we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space H. Natural applications for functions defined by power series of normal operators are given as well.
We study non-linear traces of Choquet type and Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both Choquet type and Sugeno type respectively. There exists a close relation between non-linear traces of Choquet type and majorization theory. We study trace class operators for non-linear traces of Choquet type. More generally we discuss Schatten-von Neumann $p$-class operators for non-linear traces of Choquet type. We determine when they form Banach spaces. This is an attempt of non-…
In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class of Dixmier traceable discrete pseudo-differential operators. We also present a formula for the Dixmier trace of a Dixmier traceable discrete pseudo-differential operator by using the Connes equivalence between the Wodzicki residue and the Dixmier trace.
We prove the formula for the traces of certain class of operators in bosonic and fermionic Fock spaces. Vertex operators belong to this class. Traces of vertex operators can be used for calculation of correlation functions and formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are interested in this problem. Also we show that Fredholm's minor and determinant can be expressed by such traces. We obtain a short proof of the Fredholm's formula for the solution of an integral equation. Analogue of Fredholm's formula in terms of permanents is obtained.
We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of \mathbb C^d. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.
We describe the maximal class of functions $f$ on the real line, for which the Lifshitz--Krein trace formula $\operatorname{trace}(f(A)-f(B))=\int_{\Bbb R} f'(s)\boldsymbolξ(s)\,ds$ holds for arbitrary self-adjoint operators $A$ and $B$ with $A-B$ in the trace class $\boldsymbol{S}_1$. We prove that this class of functions coincide with the class of operator Lipschitz functions.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $τ$. A closed densely defined operator $x$ affiliated with $\mathfrak{M}$ is called $τ$-measurable if there exists a number $λ\geq 0$ such that $τ\left(e^{|x|}(λ,\infty)\right)<\infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $τ$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $τ$-measurable operators. A general parallelogram law for $τ$-measura…
For Toeplitz operators on bounded symmetric domains of arbitrary rank, we define a Hilbert quotient module corresponding to partitions of length $1$ and prove that it belongs to the Macaev class ${\mathcal{L}}^{n,\infty}$. We next obtain an explicit formula for the Dixmier trace of Toeplitz commutators in terms of the underlying boundary geometry.
In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is established. Some examples of trace compatible affine spaces of operators are given.
We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.
Given B, C and W operators in the algebra L(H) of bounded linear operators on the Krein space H, the minimization problem min (BX - C)^#W(BX - C), for X in L(H), is studied when the weight W is selfadjoint. The analogous maximization and min-max problems are also considered. Complete answers to these problems and to those naturally associated to trace clase operators on Krein spaces are given.
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 …
We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix.
Some inequalities for quantum f-divergence of trace class operators in Hilbert spaces are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and chi-distance are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.
Operator model and a trace formula for pairs of unitary operators
Schmid College of Science and Technology Chapman University（查普曼大学施密德科学与技术学院）；Politecnico di Milano（米兰理工大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）
利用L. de Branges和J. Rovnyak引入的再生核希尔伯特空间理论，证明希尔伯特空间中算子对的迹公式，考虑卡拉西奥多里函数为有理函数的特殊情况，并应用于一阶离散系统理论。
We prove that every surjective isometry between the unit spheres of two trace class spaces admits a unique extension to a surjective complex linear or conjugate linear isometry between the spaces. This provides a positive solution to Tingley's problem in a new class of operator algebras.
In this paper we study Dixmier traces of powers of Hankel operators in Lorentz ideals. We extend results of Engliš-Zhang to the case of powers $p\geq 1$ and general Lorentz ideals starting from abstract extrapolation results of Gayral-Sukochev. In the special case $p=2,4,6$ we give an exact formula for the Dixmier trace. For general $p$, we give upper and lower bounds on the Dixmier trace. We also construct, for any $p$ and any Lorentz ideal, examples of non-measurable Hankel operators.
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.
We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of $\ham{D}$.
We deduce trace properties for modulation spaces of Gelfand-Shilov distributions. We use these properties to show that pseudo-differential operators with amplitudes in suitable modulation spaces, agree with pseudo-differential operators of normal type whose symbols belong to (other) modulation spaces.
Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on M, whose (complex) order is not an integer greater than or equal to -dim M, is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the L^2-operator trace on trace class operators. Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifo…
In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\boldsymbolξ$ in $L^1({\Bbb T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$ ) such that the trace formula $\operatorname{trace}(f(T_1)-f(T_0))=\int_{\Bbb T} f'(ζ)\boldsymbolξ(ζ)\,dζ$) holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.
The main result of the paper is that the Lifshits--Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that for pairs $(A_1,B_1)$ and $(A_2,B_2)$ of bounded self-adjoint operators with trace class differences $A_2-A_1$ and $B_2-B_1$, it is impossible to estimate the modulus of the trace of the difference $f(A_2,B_2)-f(A_1,B_1)$ in terms of the norm of $f$ in the Lipschitz class.
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford r…
Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later in 1988, a similar formula was obtained by Neidhardt \cite{NH} in the case of unitary operators. In this article, we give a still another proof of Koplienko-Neidhardt trace formula in the case of unitary operators by reducing the problem to a finite dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}.
The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate's trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.
Koplienko [Ko] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class $\bS_2$. A similar formula in the case of unitary operators was obtained by Neidhardt [N]. In this paper we improve their results and obtain sharp conditions under which the Koplienko--Neidhardt trace formulae hold.
Motivated by the theory of bi-singular pseudodifferential operators, we introduce a two dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra automorphism. That is, starting from an algebra equipped with an automorphism, two twisted derivations, and a twisted invariant trace, we construct an algebra of formal twisted pseudodifferential symbols and define a noncommutative residue. Also, we provide related examples.
In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$ Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)). $$ When the function $f$ is operator convex on $[0, \infty)$, the inequality is reversed.
In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\Bbb D}$, then $f(T)-f(R)\in\boldsymbol{S}_1$. The main result of the note says that there exists a function $\boldsymbolξ$ (a spectral shift function) on the unit circle ${\Bbb T}$ of class $L^1({\Bbb T})$ such that the following trace formula holds: $\operatorname{trace}(f(T…
