Abstract
This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.
Mathematics Subject Classification (2010). Primary 47A55, 47B10.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.M. Adamjan, H. Neidhardt, On the summability of the spectral shift function for pair of contractions and dissipative operators, J. Operator Theory 24 (1990), no. 1, 187–205.
V.M. Adamjan, B.S. Pavlov, Trace formula for dissipative operators, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1979, no. 2, 5–9, 118 (Russian).
A.B. Aleksandrov, V.V. Peller, Trace formulae for perturbations of class S m , J. Spectral Theory, 1 (2011), no. 1, 1–26.
A.B. Aleksandrov, V.V. Peller, D. Potapov, F.A. Sukochev, Functions of normal operators under perturbations, Adv. Math. 226 (2011), no. 6, 5216–5251.
N.A. Azamov, A.L. Carey, F.A. Sukochev, The spectral shift function and spectral flow, Comm. Math. Phys. 276 (2007), no. 1, 51–91.
N.A. Azamov, A.L. Carey, P.G. Dodds, F.A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math. 61 (2009), no. 2, 241–263.
N.A. Azamov, P.G. Dodds, F.A. Sukochev, The Krein spectral shift function in semifinite von Neumann algebras, Integral Equations Operaor Theory 55 (2006), 347–362.
M.-T. Benameur, A.L. Carey, J. Phillips, A. Rennie, F.A. Sukochev, K.P. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras. Analysis, geometry and topology of elliptic operators, 297–352, World Sci. Publ., Hackensack, NJ, 2006.
M. Sh. Birman, A.B. Pushnitski, Spectral shift function, amazing and multifaceted. Dedicated to the memory of Mark Grigorievich Krein (1907–1989), Integral Equations Operator Theory 30 (1998), no. 2, 191–199.
M.Sh. Birman, M.Z. Solomyak, Remarks on the spectral shift function, Zapiski Nauchn. Semin. LOMI 27 (1972), 33–46 (Russian). Translation: J. Soviet Math. 3 (1975), 408–419.
M.Sh. Birman, M. Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), no. 2, 131–168.
M. Sh. Birman, D.R. Yafaev, The spectral shift function. The papers of M.G. Krein and their further development, Algebra i Analiz 4 (1992), no. 5, 1–44 (Russian). Translation: St. Petersburg Math. J. 4 (1993), no. 5, 833–870.
J.-M. Bouclet, Trace formulae for relatively Hilbert-Schmidt perturbations, Asymptot. Anal. 32 (2002), 257–291.
R.W. Carey, J.D. Pincus, Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977), 153–218.
M. Caspers, S. Montgomery-Smith, D. Potapov, F. Sukochev, The best constants for operator Lipschitz functions on Schatten classes, preprint.
A.H. Chamseddine, A. Connes, The spectral action principle, Comm. Math. Phys. 186 (1997), 731–750.
Yu.L. Daleckii, S.G. Krein, Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations, (Russian) Vorone. Gos. Univ. Trudy Sem. Funkcional. Anal. 1956 (1956), no. 1, 81–105.
K. Dykema, A. Skripka, Higher order spectral shift, J. Funct. Anal. 257 (2009), 1092–1132.
K. Dykema, A. Skripka, Perturbation formulas for traces on normed ideals, Comm. Math. Phys. 325 (2014), no 3, 1107–1138.
Y.B. Farforovskaya, An estimate of the nearness of the spectral decompositions of self-adjoint operators in the Kantorovič–Rubinštein metric, Vestnik Leningrad Univ. 22 (1967), no. 19, 155–156.
Y.B. Farforovskaya, The connection of the Kantorovič–Rubinštein metric for spectral resolutions of self-adjoint operators with functions of operators, Vestnik Leningrad Univ. 23 (1968), no. 19, 94–97.
Y.B. Farforovskaya, An example of a Lipschitz function of self-adjoint operators with non-nuclear difference under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 146–153.
F. Gesztesy, A. Pushnitski, B. Simon, On the Koplienko spectral shift function, I. Basics, Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107.
E. Kissin, D. Potapov, V. Shulman, F. Sukochev, Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations, Proc. London Math. Soc. 105 (2012), no. 4, 661–702.
L.S. Koplienko, Local conditions for the existence of the function of spectral shift. Investigations on linear operators and the theory of functions, VIII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 73 (1977), 102–117 (Russian). Translation: J. Soviet Math. 34 (1986), no. 6, 2080–2090.
L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984), 62–71 (Russian). Translation: Siberian Math. J. 25 (1984), 735–743.
M.G. Krein, On a trace formula in perturbation theory, Matem. Sbornik 33 (1953), 597–626 (Russian).
M.G. Krein, On the perturbation determinant and the trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268–271 (Russian). Translation: Soviet Math. Dokl. 3 (1962), 707–710.
M.G. Krein, Some new studies in the theory of perturbations of self-adjoint operators, First Math. Summer School, Part I, 1964, pp. 103–187, Izdat. “Naukova Dumka”, Kiev (Russian). Translation: Topics in differential and integral equations and operator theory, Birkhäuser Verlag, Basel, 1983, pp. 107–172.
M.G. Krein, Perturbation determinants and a trace formula for some classes of pairs of operators, J. Operator Theory 17 (1987), no. 1, 129–187 (Russian).
H. Langer, Eine Erweiterung der Spurformel der Störungstheorie, Math. Nachr. 30, 123–135 (1965) (German).
I.M. Lifshits, On a problem of the theory of perturbations connected with quantum statistics, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1 (47), 171–180 (Russian).
S. Lord, F. Sukochev, D. Zanin, Singular Traces, de Gruyter Studies in Mathematics, 46, Walter de Gruyter & Co., Berlin, 2012.
K.A. Makarov, A. Skripka, M. Zinchenko, On perturbation determinant for antidissipative operators, preprint.
H. Neidhardt, Scattering matrix and spectral shift of the nuclear dissipative scattering theory. Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985), 237–250, Oper. Theory Adv. Appl., 24, Birkhäuser, Basel, 1987.
H. Neidhardt, Scattering matrix and spectral shift of the nuclear dissipative scattering theory. II. J. Operator Theory 19 (1988), no. 1, 43–62.
H. Neidhardt, Spectral shift function and Hilbert–Schmidt perturbation: extensions of some work of L.S. Koplienko, Math. Nachr. 138 (1988), 7–25.
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, 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.
V.V. Peller, The behavior of functions of operators under perturbations. A glimpse at Hilbert space operators, 287–324, Oper. Theory Adv. Appl., 207, Birkhäuser Verlag, Basel, 2010.
G. Pisier, Q. Xu, Noncommutative L p -spaces. Handbook of the Geometry of Banach spaces, 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.
D. Potapov, A. Skripka, F. Sukochev, Spectral shift function of higher order, Invent. Math., 193 (2013), no. 3, 501–538.
D. Potapov, A. Skripka, F. Sukochev, On Hilbert–Schmidt compatibility, Oper. Matrices, 7 (2013), no. 1, 1–34.
D. Potapov, A. Skripka, F. Sukochev, Higher order spectral shift for contractions, Proc. London Math. Soc. 108 (2014), no 3, 327–349.
D. Potapov, F. Sukochev, Operator-Lipschitz functions in Schatten–von Neumann classes, Acta Math., 207 (2011), 375–389.
D. Potapov, F. Sukochev, Koplienko spectral shift function on the unit circle, Comm. Math. Phys., 309 (2012), 693–702.
D. Potapov, F. Sukochev, D. Zanin, Krein’s trace theorem revisited, J. Spectral Theory, in press.
A.V. Rybkin, The spectral shift function for a dissipative and a selfadjoint operator, and trace formulas for resonances, Mat. Sb. (N.S.) 125(167) (1984), no. 3, 420–430 (Russian).
A.V. Rybkin, The spectral shift function, the characteristic function of a contraction and a generalized integral, Mat. Sb. 185 (1994), no. 10, 91–144 (Russian). Translation: Russian Acad. Sci. Sb. Math. 83 (1995), no. 1, 237–281.
A.V. Rybkin, On A-integrability of the spectral shift function of unitary operators arising in the Lax–Phillips scattering theory, Duke Math. J. 83 (1996), no. 3, 683–699.
B. Simon, Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.
A. Skripka, Trace inequalities and spectral shift, Oper. Matrices 3 (2009), no. 2, 241–260.
A. Skripka, Higher order spectral shift, II. Unbounded case, Indiana Univ. Math. J. 59 (2010), no. 2, 691–706.
A. Skripka, Multiple operator integrals and spectral shift, Illinois J. Math., 55 (2011), no. 1, 305–324.
A. Skripka, Asymptotic expansions for trace functionals, J. Funct. Anal. 266 (2014), no 5, 2845–2866.
B. Sz.-Nagy, C. FoiaĹź, Harmonic analysis of operators on Hilbert space. Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London, 1970.
W.D. van Suijlekom, Perturbations and operator trace functions, J. Funct. Anal. 260 (2011), no. 8, 2483–2496.
D.R. Yafaev, Mathematical scattering theory: general theory, Providence, R.I., AMS, 1992.
D.R. Yafaev, The Schrödinger operator: perturbation determinants, the spectral shift function, trace identities, and more, Funktsional. Anal. i Prilozhen. 41 (2007), no. 3, 60–83 (Russian). Translation: Funct. Anal. Appl. 41 (2007), no. 3, 217–236.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Skripka, A. (2014). Taylor Approximations of Operator Functions. In: Ball, J., Dritschel, M., ter Elst, A., Portal, P., Potapov, D. (eds) Operator Theory in Harmonic and Non-commutative Analysis. Operator Theory: Advances and Applications, vol 240. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06266-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-06266-2_12
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-06265-5
Online ISBN: 978-3-319-06266-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)