arXiv subjects
Publications and source records attributed to Anna Skripka.
In this note, we summarize known results and open questions on the existence of isometric embeddings between different Schatten classes as well as obtain a new non-embeddability result using a novel method. We also provide a brief overview of the relevant methods.
We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of Schatten--von Neumann perturbations of (possibly unbounded) self-adjoint operators.
We establish higher order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator of the contractions and enlarging the set of admissible functions. We also derive higher order trace formulas for maximal dissipative operators under relaxed assumptions and new simplified trace formulas for unitary and resolvent comparable self-adjoint operators. The respective spectral shift measures are absolutely continuous and, in the case of contractions, the set of admissible functions for the $n$th order trace formula on the unit circle includes the Besov class $B^n_{\infty, 1}(\T)$. Both aforementioned properties are new in the mentioned generality.
We establish estimates and representations for the remainders of Taylor approximations of the spectral action functional $V\mapstoτ(f(H_0+V))$ on bounded self-adjoint perturbations, where $H_0$ is a self-adjoint operator with $τ$-compact resolvent in a semifinite von Neumann algebra and $f$ belongs to a broad set of compactly supported functions including $n$-times differentiable functions with bounded $n$-th derivative. Our results significantly extend analogous results in \cite{SkAnJOT}, where $f$ was assumed to be compactly supported and $(n+1)$-times continuously differentiable. If, in addition, the resolvent of $H_0$ belongs to the noncommutative $L^n$-space, stronger estimates are derived and extended to noncompactly supported functions with suitable decay at infinity.
Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations $V$ and settles earlier attempts to encompass general perturbations with Schatten-von Neumman difference of resolvents, which led to more complicated trace formulas for more restrictive sets of functions $f$ and to analogs of spectral shift functions lacking real-valuedness and/or expected degree of uniqueness. Our proof builds on a general change of variables method derived in this paper and significantly refining those appearing in \cite{vNS21,PSS15,S17} with respect to several parameters at once.
We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $\mathcal{S}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $V\in\mathcal{S}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.
We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet $\mathcal{S}^p$-differentiable and $n$ times Gâteaux $\mathcal{S}^p$-differentiable at every self-adjoint operator. We also prove that if $f\in B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet $\mathcal{S}^q$-differentiable, $1\le q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of [10] to arbitrary $n$ and unbounded operators as well as substantially extend the results of [2,4,19] on higher order $\mathcal{S}^p$-differentiability of $f$ in a certain Wiener class, Gâteaux $\mathcal{S}^2$-differentiability of $f\in C^n(\mathbb{R})$ with $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, and Gâteaux $\mathcal{S}^q$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$. As an application, we extend $\mathcal{S}^p$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.
Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\in C^n(\mathbb{R})$. We establish that $φ(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\ldots,n$, are bounded. As an application of the second order $\Sc^2$-differentiability, we extend the Koplienko trace formula from the Besov class $B_{\infty1}^2(\R)$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.
We prove a version of Holder's inequality with a constant for p-th roots of symmetric operator spaces of operators affiliated to a semifinite von Neumann algebra factor, and with constant equal to 1 for strongly symmetric operator spaces.
For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.
We obtain Taylor approximations for functionals $V\mapsto Tr(f(H_0+V))$ defined on the bounded self-adjoint operators, where $H_0$ is a self-adjoint operator with compact resolvent and $f$ is a sufficiently nice scalar function, relaxing assumptions on the operators made in [17], and derive estimates and representations for the remainders of these approximations.
This paper extends the trace formulas of [5] with perturbations in normed ideals of $B(H)$ to multivariate functions of commuting contractions admitting a dilation to commuting normal contractions.
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 show that every self--adjoint matrix B of trace 0 can be realized as B=T+T^* for a nilpotent matrix T of norm no greater than K times the norm of B, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self--adjoint n-by-n matrix of trace 0, then there is a unitary matrix V=XU_n, where X is an n-by-n permutation matrix and U_n is the n-by-n Fourier matrix, such that the upper triangular part, T, of the conjugate V^*DV of D has norm no greater than K times the norm of D. This matrix T is a strictly upper triangular Toeplitz matrix such that T+T^*=V^*DV. We apply this and related results to give partial answers to questions about real parts of quasinilpotent elements in finite von Neumann algebras.
This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.
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.
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.
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.