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Double Operator Integrals and Submajorization

Published online by Cambridge University Press:  12 May 2010

D. Potapov  and
F. Sukochev
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Abstract

We present a user-friendly version of a double operator integration theory which stillretains a capacity for many useful applications. Using recent results from the lattertheory applied in noncommutative geometry, we derive applications to analogues of theclassical Heinz inequality, a simplified proof of a famous inequality ofBirman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods aresufficiently strong to treat these inequalities in the setting of symmetric operator normsin general semifinite von Neumann algebras.

Keywords

double operator integrationunitarily invariant norm inequalitiesnoncommutative L p -spaces
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 317 - 339
Copyright
© EDP Sciences, 2010

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References

Ando, T.. Comparison of norms |||f(A) − f(B) ||| and |||f(|A − B|) |||. Math. Z., 197 (1988), No. 3, 403–409. CrossRefGoogle Scholar
Azamov, N. A., Carey, A. L., Dodds, P. G., Sukochev, F. A.. Operator integrals, spectral shift, and spectral flow . Canad. J. Math., 61 (2009), No. 2, 241–263.CrossRefGoogle Scholar
Birman, M. Š., Koplienko, L. S., Solomjak, M. Z.. Estimates of the spectrum of a difference of fractional powers of selfadjoint operators . Izv. Vysš. Učebn. Zaved. Matematika, 154 (1975), No. 3, 3–10.Google Scholar
Birman, M. S., Solomyak, M. Z.. Double Stieltjes operator integrals . Problemy Mat. Fiz., (1966), No. 1, 33–67 (Russian). Google Scholar
Birman, M. S., Solomyak, M. Z.. Double Stieltjes operator integrals, II . Problemy Mat. Fiz., (1967), No. 2, 26–60 (Russian). Google Scholar
Birman, M. S., Solomyak, M. Z.. Double Stieltjes operator integrals, III . Problemy Mat. Fiz., (1973), No. 6, 27–53 (Russian). Google Scholar
Daleckiĭ, Yu. L., Kreĭn, S.G.. Formulas of differentiation according to a parameter of functions of Hermitian operators . Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 13–16.Google Scholar
Daleckiĭ, Yu. L., Kreĭn, S.G.. Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations . Voronež. Gos. Univ. Trudy Sem. Funkcional. Anal., 1 (1956), 81–105.Google Scholar
Carey, A. L., Potapov, D. S., Sukochev, F. A.. Spectral flow is the integral of one forms on Banach manifolds of self adjoint Fredholm operators . Adv. Math, 222 (2009), 1809–1849.CrossRefGoogle Scholar
Connes, A., Moscovici, H.. Transgression du caractère de Chern et cohomologie cyclique . C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), No. 18, 913–918.Google Scholar
Dodds, P. G., Dodds, T. K.. On a submajorization inequality of T. Ando. Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., 75, (1995), 113–131. Google Scholar
Dodds, P. G., Sukochev, F.A.. Submajorisation inequalities for convex and concave functions of sums of measurable operators . Positivity, 13 (2009), No. 1, 107–124. CrossRefGoogle Scholar
I.C. Gohberg. M.G. Kreĭn. Introduction to the theory of linear nonselfadjoint operators. Translations of Mathematical Monographs, Providence, R.I., AMS, 18, 1969.
Kalton, N.J., Sukochev, F.A. Symmetric norms and spaces of operators . J. Reine Angew. Math., 621 (2008), 81–121. Google Scholar
H. Kosaki. Positive definiteness of functions with applications to operator norm inequalities. Preprint, 2009.
de Pagter, B., Sukochev, F. A.. Differentiation of operator functions in non-commutative Lp-spaces . J. Funct. Anal., 212 (2004), No. 1, 28–75.CrossRefGoogle Scholar
de Pagter, B., Sukochev, F. A.. Commutator estimates and R-flows in non-commutative operator spaces . Proc. Edinb. Math. Soc., 50 (2007), No. 2, 293–324.CrossRefGoogle Scholar
de Pagter, B., Sukochev, F. A., Witvliet, H.. Double operator integrals . J. Funct. Anal., 192 (2002), No. 1, 52–111.CrossRefGoogle Scholar
Gesztesy, F., Pushnitski, A., Simon, B.. On the Koplienko spectral shift function. I. Basics . Zh. Mat. Fiz. Anal. Geom., 4 (2008), No. 1, 63–107.Google Scholar
Heinz, E.. Beiträge zur Störungstheorie der Spektralzerlegung . Math. Ann., 123 (1951), 415–438.CrossRefGoogle Scholar
A. McIntosh. Heinz inequalities and perturbation of spectral families. Macquarie Mathematical Reports, 79–0006 (1979).
G. Pisier, Q. Xu. Non-commutative L p -spaces. Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459–1517.
Potapov, D., Sukochev, F.. Lipschitz and commutator estimates in symmetric operator spaces. J. Operator Theory, 59 (2008), No. 1, 211–234. Google Scholar
Potapov, D., Sukochev, F.. Unbounded Fredholm modules and double operator integrals . J. Reine Angew. Math., 626 (2009), 159–185.Google Scholar
Segal, I.E.. A non-commutative extension of abstract integration . Annals of Mathematics, 57 (1953), 401–457.CrossRefGoogle Scholar
B. Simon. Trace ideals and their applications. Mathematical Surveys and Monographs, AMS, Providence, RI, 120 (2005).
Sukochev, F. A., Chilin, V. I.. The triangle inequality for operators that are measurable with respect to Hardy-Littlewood order . Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, (1988), No. 4, 44–50 (Russian). Google Scholar
Sukochev, F. A., Chilin, V. I.. Symmetric spaces over semifinite von Neumann algebras . Soviet Math. Dokl., 42 (1991), No. 1, 97–101 (Russian). Google Scholar
Sukochev, F. A., Chilin, V. I.. Weak convergence in non-commutative symmetric spaces . J. Operator Theory, 31 (1994), No. 1, 35–65.Google Scholar
von Neumann, J.. Some matrix inequalities and metrization of matric-space . Rev. Tomsk Univ., 1 (1937), 286–300.Google Scholar