Illinois Journal of Mathematics

Multiple operator integrals and spectral shift

Anna Skripka

Full-text: Open access

Abstract

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.

Article information

Source
Illinois J. Math., Volume 55, Number 1 (2011), 305-324.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1355927038

Digital Object Identifier
doi:10.1215/ijm/1355927038

Mathematical Reviews number (MathSciNet)
MR3006690

Zentralblatt MATH identifier
1264.47019

Subjects
Primary: 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 46L52: Noncommutative function spaces

Citation

Skripka, Anna. Multiple operator integrals and spectral shift. Illinois J. Math. 55 (2011), no. 1, 305--324. doi:10.1215/ijm/1355927038. https://projecteuclid.org/euclid.ijm/1355927038


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References

  • N. A. Azamov, A. L. Carey, P. G. Dodds and F. A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math. 61 (2009), 241–263.
  • N. A. Azamov, P. G. Dodds and F. A. Sukochev, The Krein spectral shift function in semifinite von Neumann algebras, Integral Equations Operator Theory 55 (2006), 347–362.
  • M. S. Birman and M. Z. Solomyak, Tensor product of a finite number of spectral measures is always a spectral measure, Integral Equations Operator Theory 24 (1996), 179–187.
  • R. W. Carey and J. D. Pincus, Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977), 153–218.
  • R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, vol. 303, Springer-Verlag, Berlin, 1993.
  • K. Dykema and A. Skripka, Higher order spectral shift, J. Funct. Anal. 257 (2009), 1092–1132.
  • L. S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984), 62–71 (Russian). Translation: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984), 735–743.
  • M. G. Krein, On a trace formula in perturbation theory, Matem. Sbornik 33 (1953), 597–626 (Russian).
  • I. M. Lifshits, On a problem of the theory of perturbations connected with quantum statistics, Uspehi Matem. Nauk 7 (1952), 171–180 (Russian).
  • H. Neidhardt, Spectral shift function and Hilbert–Schmidt perturbation: Extensions of some work of L. S. Koplienko, Math. Nachr. 138 (1988), 7–25.
  • B. S. Pavlov, On multidimensional operator integrals, Problems of mathematical analysis, vol. 2, Leningrad State Univ. 1969, pp. 99–121 (Russian).
  • V. V. Peller, Hankel operators in the perturbation theory of unbounded self-adjoint operators, Analysis and partial differential equations, Lecture Notes in Pure and Applied Mathematics, vol. 122, Dekker, New York, 1990, pp. 529–544.
  • V. V. Peller, An extension of the Koplienko–Neidhardt trace formulae, J. Funct. Anal. 221 (2005), 456–481.
  • V. V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 223 (2006), 515–544.
  • A. Skripka, Trace inequalities and spectral shift, Oper. Matrices 3 (2009), 241–260.
  • A. Skripka, Higher order spectral shift, II. Unbounded case, Indiana Univ. Math. J. 59 (2010), 691–706.