Publications and Preprints

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Research supported in part by NSF DMS grants 0900870, 1249186, 1500704, 1554456.

[43] "Lower bounds for the tracial seminorm of random Schur multipliers and Gaussian chaoses"
[42] "Higher-order spectral shift function for resolvent comparable perturbations" (w/ T. van Nuland), Journal of Operator Theory, to appear.
Results: existence and integrability of the higher-order spectral shift functions satisfying the optimal trace formulas for perturbations entailing a Schatten class difference of the resolvents; general change of variables method for multilinear operator integrals.
[41] "Approximation of the spectral action functional in the case of τ-compact resolvents" (w/ A. Chattopadhyay, C. Pradhan), Integral Equations and Operator Theory, 95 (2023), no. 3, paper no. 20.
Results: perturbation theory of the spectral action functional in the semi-finite von Neumann algebra setting, significant relaxation of previous smoothness assumptions imposed on admissible functions f in the case of τ-compact resolvents and, in addition, removal of the compact support assumption on f in the case of τ-Schatten-von Neumann resolvents.
[40] "Lipschitz-type bounds for functions of operators with noncompact perturbations", Operator Theory: Advances and Applications, 290, Birkhäuser, Basel, 2023, 345-358.
Results: Lipschitzness of operator functions in the setting of noncompact perturbations that arise in problems of mathematical physics and noncommutative geometry; survey + new results.
[39] "Spectral shift for relative Schatten class perturbations" (w/ T. van Nuland), Journal of Spectral Theory, 12 (2022), no. 4, 1347-1382.
Results: existence of a real-valued, unique up to a polynomial term, spectral shift function that satisfies the best higher order trace formula for a significantly enlarged set of functions. Done for a perturbation whose product with the resolvent of an initial operator is an element of a Schatten class.
[38] "Sharpening bounds for multilinear Schur multipliers", La Matematica, (2022), no. 1, 167-185.
Results: survey of major estimates for norms and traces of multilinear Schur multipliers as well as open questions and partial results on delicate lower bounds, which were scattered in the literature or not published.
[37] Editor's pick "Lipschitz estimates for functions of Dirac and Schrödinger operators", Journal of Mathematical Physics, 62, 013506 (2021), no. 1, 28 pp.
Results: characterization of functions satisfying a Lipschitz-type bound for resolvent comparable perturbations with respect to the Schatten p-norm, p>1; explicit dependence on the Lipschitz seminorm and decay parameters of the respective scalar functions and, in the case of Dirac and Schrödinger operators, on the L^p- or l^p(L²)-norm of the potential; long-range and random potentials.
[36] "MSE bounds for estimators of matrix functions", Linear Algebra and its Applications, 609 (2021), 231-252.
Results: optimal bounds for Schatten norms of mean squared errors of plug-in estimators of matrix functions; applications to estimators of covariance matrix functions with reduction of dimensional dependence of a sample size.
[35] "Differentiability of operator functions in Schatten norms" (w/ C. Le Merdy), Journal of the Institute of Mathematics of Jussieu, 19 (2020), no. 6, 1993-2016.
Results: characterization of an arbitrary order continuous Fréchet differentiability; Gâteaux differentiability for a very broad set of functions.
[34] "Tracial bounds for multilinear Schur multipliers", Linear Algebra and its Applications, 590 (2020), 62-84.
Results: new types of two-sided bounds for traces of multilinear Schur multipliers; characterization of tracial positivity on symmetric tuples in case of mildly sparse symbols.
[33] "On uniqueness of higher order spectral shift functions" (w/ M. Zinchenko), Studia Mathematica, 251 (2020), no. 2, 207-218.
Results: non-uniqueness of the first order and uniqueness of higher order approximations; a nonlinear inverse problem.
[32] Featured article "Untangling noncommutativity with operator integrals", Notices of the American Mathematical Society, 67 (2020), no. 1, 45-55.
[31] Book "Multilinear Operator Integrals: Theory and Applications" (w/ A. Tomskova), Lecture Notes in Mathematics, 2250, Springer, 2019, XI+192 pp.
[30] "Stability and uniqueness properties of Taylor approximations of matrix functions" (w/ M. Zinchenko), Linear Algebra and its Applications, 582 (2019), 218-236.
Results: lower and upper bounds for remainders; applications to accuracy of approximations and spectral graph theory.
[29] Fall sectional sampler "Operator Integrals in Theory and Applications", Notices of the American Mathematical Society, 66 (2019), no. 10, 1699-1700.
[28] "Higher order S²-differentiability and application to Koplienko trace formula" (w/ C. Coine, C. Le Merdy, F. Sukochev), Journal of Functional Analysis, 276 (2019), no. 10, 3170-3204.
Results: existence of higher order operator derivatives in S² for a broad set of functions; trace formula for functions with second divided difference admitting a Hilbert space factorization.
[27] "Taylor asymptotics of spectral action functionals", Journal of Operator Theory, 80 (2018), no. 1, 113-124.
Results: asymptotic expansion in the case of operators with τ-compact resolvents (without summability restrictions on resolvents and perturbations), with remainder estimate similar to the one in [12].
[26] "Trace formulas for relative Schatten class perturbations" (w/ A. Chattopadhyay), Journal of Functional Analysis, 274 (2018), 3377-3410.
Results: Taylor approximation of operator functions, provided the product of a perturbation and the resolvent of an initial operator is an element of a Schatten class; applications to differential operators.
[25] "Multilinear Schur multipliers and applications to operator Taylor remainders" (w/ D. Potapov, F. Sukochev, A. Tomskova), Advances in Mathematics, 320 (2017), 1063-1098.
Results: dimension dependent estimates from below for multilinear Schur multipliers; affirmative and negative results on summability of operator Taylor remainders; sharp conditions.
[24] "Estimates and trace formulas for unitary and resolvent comparable perturbations", Advances in Mathematics, 311 (2017), 481-509.
Results: extension of estimates obtained in [13] from polynomials to trigonometric polynomials and Fourier series; extension of corollaries of these estimates obtained in [20,22].

[23] "On positivity of spectral shift functions", Linear Algebra and its Applications, 523 (2017), 118-130.
Results: partial results on sign-definiteness of higher order spectral shift functions.
[22] "Functions of unitary operators: derivatives and trace formulas" (w/ D. Potapov, F. Sukochev), Journal of Functional Analysis, 270 (2016), 2048-2072.
Results: formulas for derivatives of operator functions along multiplicative paths of unitaries and respective trace formulas.

[21] "Hölder's inequality for roots of symmetric operator spaces" (w/ K. Dykema), Studia Mathematica, 228 (2015), 47-54.
Results: Hölder's inequality for roots of symmetric operator spaces corresponding to semifinite von Neumann factors.

[20] "Trace formulas for resolvent comparable operators" (w/ D. Potapov, F. Sukochev), Advances in Mathematics, 272 (2015), 630-651.
Results: non-Taylor approximation of operator functions, provided the difference of resolvents is in a Schatten class; applications to differential operators.

[19] "On a perturbation determinant for accumulative operators" (w/ K. A. Makarov, M. Zinchenko), Integral Equations and Operator Theory, 81 (2015), no. 3, 301-317.
Results: an exponential representation for the perturbation determinant, a respective trace formula; the spectral shift function is not in L¹_w,0.

[18] "Trace formulas for multivariate operator functions", Integral Equations and Operator Theory, 81 (2015), no. 4, 559-580.
Results: extension of results of [14] to the multivariable case.

[17] "Taylor approximations of operator functions", Operator Theory: Advances and Applications, 240, Birkhäuser, Basel, 2014, 243-256.
Results: a survey of a 70 years long development of the subject.

[16] "Asymptotic expansions for trace functionals", Journal of Functional Analysis, 266 (2014), no. 5, 2845-2866.
Results: Taylor approximation of the trace of an operator function in the case of an unsummable perturbation, provided the initial operator has compact resolvent.

[15] "Upper triangular Toeplitz matrices and real parts of quasinilpotent operators" (w/ K. Dykema, J. Fang), Indiana University Mathematics Journal, 63 (2014), no. 1, 53-75.
Results: a self-adjoint matrix A of trace zero is a real part of a nilpotent matrix Q, where ||Q|| is controlled by ||A|| independently of the dimension; realization of some zero trace self-adjoint elements in a II₁ factor as a real part of a quasinilpotent.

[14] "Perturbation formulas for traces on normed ideals" (w/ K. Dykema), Communications in Mathematical Physics, 325 (2014), no. 3, 1107-1138.
Results: first and second order trace formulas for a symmetric operator ideal with a positive bounded trace; non-absolutely continuous spectral shift measure.
A definition erratum, Comm. Math. Phys., 340 (2015), no. 2, 865.

[13] "Spectral shift function of higher order for contractions" (w/ D. Potapov, F. Sukochev), Proceedings of the London Mathematical Society, 108 (2014), no. 3, 327-349.
Results: estimates for Schatten norms of multiple operator integrals in the case of contractions and respective trace formulas.

[12] "Spectral shift function of higher order" (w/ D. Potapov, F. Sukochev), Inventiones Mathematicae, 193 (2013), no. 3, 501-538.
Results: new approach to multiple operator integrals and estimates for their Schatten norms in the self-adjoint case; resolution of Koplienko's conjecture of 1984.

[11] "On Hilbert-Schmidt compatibility" (w/ D. Potapov, F. Sukochev), Operators and Matrices, 7 (2013), no. 1, 1-34.
Results: trace formulas for Hilbert-Schmidt compatible operators that are simpler than those in [20].

[10] "On single commutators in II₁ factors" (w/ K. Dykema), Proceedings of the American Mathematical Society, 140 (2012), no. 3, 931-940.
Results: representation of new zero trace elements in a II₁ factor as single commutators.

[9] "Multiple operator integrals and spectral shift", Illinois Journal of Mathematics, 55 (2011), no. 1, 305-324.
Results: relation of higher order spectral shift functions to multiple operator integrals.

[8] "Higher order spectral shift, II. Unbounded case", Indiana University Mathematics Journal, 59 (2010), no. 2, 691-706.
Results: extension of results of [5] to unbounded operators.

[7] "On the centralizer of a one-parameter representation", Operator theory live, Theta Series in Advanced Mathematics, 12, Theta, Bucharest, 2010, 179-187.
Results: Stieltjes-type integral representation for operators commuting with a one-parameter representation on a Banach space.

[6] "Some applications of the perturbation determinant in finite von Neumann algebras" (w/ K. A. Makarov), Canadian Journal of Mathematics, 62 (2010), no. 1, 133-156.
Results: introduction of the perturbation determinant based on the de la Harpe-Skandalis determinant; generalizations of the Birman-Solomyak spectral averaging formula.

[5] "Higher order spectral shift" (w/ K. Dykema), Journal of Functional Analysis, 257 (2009), 1092-1132.
Results: higher order trace formulas under the assumption V be in S²; recursive formulas for higher order spectral shift functions.

[4] "Trace inequalities and spectral shift", Operators and Matrices, 3 (2009), no. 2, 241-260.
Results: monotonicity and concavity of traces of operator functions.

[3] "On properties of the ξ-function in semi-finite von Neumann algebras", Integral Equations and Operator Theory, 62 (2008), no. 2, 247-267.
Results: extension of results of [1] to semifinite von Neumann algebras.

[2] "A spectral integral representation for decomposable operators", Mathematische Nachrichten, 281 (2008), no. 3, 1-10.
Results: Lebesgue-type integral representation for operators commuting with a spectral measure.

[1] "The Birman-Schwinger principle in von Neumann algebras of finite type" (w/ V. Kostrykin, K. A. Makarov), Journal of Functional Analysis, 247 (2007), 492-508.
Results: an analog of the Birman-Schwinger eigenvalue counting principle and the Birman-Krein formula in finite von Neumann algebras.