* Research supported in part by NSF DMS grants 0900870, 1249186, 1500704.
[32] *"Multilinear operator integrals: development and applications", (w/ A. Tomskova), in progress.
[31] *"Stability of a uniqueness problem", (w/ M. Zinchenko), in progress.
[30] *"Differentiability of operator functions in Schatten norms", (w/ C. Le Merdy), in progress.
[29] *"Higher order S^{2}-differentiability and application to Koplienko trace formula", (w/ C. Coine, C. Le Merdy, F. Sukochev), preprint.
Results: existence of higher order operator derivatives in S^{2} for and extension of trace formulas to a very broad set of functions.
[28] *"A uniqueness property of spectral sum approximations", (w/ M. Zinchenko), preprint.
Results: non-uniqueness of the first order and uniqueness of higher order approximations. This is a nonlinear inverse problem.
[27] *"Trace formulas for relative Schatten class perturbations", (w/ Arup 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.
[26] *"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.
[25] *"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, where the estimate is like the one in [12].
[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.
[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 L^{1}_{w,0}.
[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 II_{1} 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 II_{1} 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 II_{1} 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.
[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 S^{2}; 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.
[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.
"Trace formulae in finite von Neumann algebras", PhD Thesis, University of Missouri, 2007, 95 pp., ISBN: 978-0549-72651-7.
Results: the ξ-index, Birman-Schwinger principle and its consequences, spectral averaging in finite von Neumann algebras.