Fuglede commutations modulo Lorentz ideals

Xia, Jingbo

doi:10.1090/tran/9058

Fuglede commutations modulo Lorentz ideals
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by Jingbo Xia;

Trans. Amer. Math. Soc. 377 (2024), 249-269

DOI: https://doi.org/10.1090/tran/9058

Published electronically: October 19, 2023

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Abstract:

We examine Fuglede commutation properties, particularly those in the context of Lorentz ideals, from a new perspective. We also show that Fuglede commutation property fails for a number of ideals.

References

A. Abdessemed and E. B. Davies, Some commutator estimates in the Schatten classes, J. London Math. Soc. (2) 39 (1989), no. 2, 299–308. MR 991663, DOI 10.1112/jlms/s2-39.2.299
Hari Bercovici and Dan Voiculescu, The analogue of Kuroda’s theorem for $n$-tuples, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988) Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 57–60. MR 1038331
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Guy David and Dan Voiculescu, $s$-numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions, J. Funct. Anal. 94 (1990), no. 1, 14–26. MR 1077543, DOI 10.1016/0022-1236(90)90026-H
Jacques Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107–A1108 (French). MR 196508
Quanlei Fang and Jingbo Xia, Schatten class membership of Hankel operators on the unit sphere, J. Funct. Anal. 257 (2009), no. 10, 3082–3134. MR 2568686, DOI 10.1016/j.jfa.2009.05.006
Quanlei Fang and Jingbo Xia, A local inequality for Hankel operators on the sphere and its application, J. Funct. Anal. 266 (2014), no. 2, 876–930. MR 3132732, DOI 10.1016/j.jfa.2013.10.003
Quanlei Fang and Jingbo Xia, On the membership of Hankel operators in a class of Lorentz ideals, J. Funct. Anal. 267 (2014), no. 4, 1137–1187. MR 3217060, DOI 10.1016/j.jfa.2014.05.010
Quanlei Fang and Jingbo Xia, Best approximations in a class of Lorentz ideals, Complex Anal. Oper. Theory 16 (2022), no. 4, Paper No. 51, 25. MR 4403190, DOI 10.1007/s11785-022-01220-z
Bent Fuglede, A commutativity theorem for normal operators, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 35–40. MR 32944, DOI 10.1073/pnas.36.1.35
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, RI, 1969. Translated from the Russian by A. Feinstein. MR 246142, DOI 10.1090/mmono/018
J. William Helton and Roger E. Howe, Traces of commutators of integral operators, Acta Math. 135 (1975), no. 3-4, 271–305. MR 438188, DOI 10.1007/BF02392022
Liangying Jiang, Yi Wang, and Jingbo Xia, Toeplitz operators associated with measures and the Dixmier trace on the Hardy space, Complex Anal. Oper. Theory 14 (2020), no. 2, Paper No. 30, 58. MR 4067304, DOI 10.1007/s11785-020-00988-2
B. E. Johnson and J. P. Williams, The range of a normal derivation, Pacific J. Math. 58 (1975), no. 1, 105–122. MR 380490, DOI 10.2140/pjm.1975.58.105
Edward Kissin and Victor S. Shulman, Classes of operator-smooth functions. I. Operator-Lipschitz functions, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 1, 151–173. MR 2117717, DOI 10.1017/S0013091503000178
Edward Kissin and Victor S. Shulman, Classes of operator-smooth functions. III. Stable functions and Fuglede ideals, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 1, 175–197. MR 2117718, DOI 10.1017/S001309150300018X
Steven Lord, Fedor Sukochev, and Dmitriy Zanin, Singular traces, De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013. Theory and applications. MR 3099777
Victor Shulman, Some remarks on the Fuglede-Weiss theorem, Bull. London Math. Soc. 28 (1996), no. 4, 385–392. MR 1384827, DOI 10.1112/blms/28.4.385
Victor Shulman and Lyudmila Turowska, Operator synthesis. II. Individual synthesis and linear operator equations, J. Reine Angew. Math. 590 (2006), 143–187. MR 2208132, DOI 10.1515/CRELLE.2006.007
Dan Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), no. 1, 3–37. MR 553861
Gary Weiss, The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II, J. Operator Theory 5 (1981), no. 1, 3–16. MR 613042
Gary Weiss, The Fuglede commutativity theorem modulo operator ideals, Proc. Amer. Math. Soc. 83 (1981), no. 1, 113–118. MR 619994, DOI 10.1090/S0002-9939-1981-0619994-2
Jingbo Xia, Singular integral operators associated with measures of varying density, J. Operator Theory 49 (2003), no. 2, 311–324. MR 1991741