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RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
Maps preserving absolute continuity of positive operators
Tamás Titkos, titkos@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Zsigmond Tarcsay, György Pál Gehér
Motivated by their measure theoretic analogues, Ando introduced in [1] the notions of absolute
continuity and singularity of positive operators, and proved a Lebesgue-type decomposition
theorem. Since then, similar results have been proved in more general contexts [3, 4, 5, 7, 8],
just to mention a few. Molnár in [6] described the structure of bijective maps on the cone of
positive operators that preserve the Lebesgue decomposition in both directions. It turned out that
the cone is quite rigid in the sense that these maps can be always written in the form A → SAS∗
with a bounded, invertible, linear- or conjugate linear operator S. A natural question arises: is
it possible to weaken the preserver property, and to characterize those bijections that preserve
absolute continuity only? The aim of this talk to answer this question in the affirmative.
References
[1] ANDO, T. Lebesgue-type decomposition of positive operators. Acta Sci. Math. (Szeged)
38 (1976), no. 3-4, 253–260.
[2] GY.P. GEHÉR, ZS. TARCSAY, T. TITKOS, Maps preserving absolute continuity and sin-
gularity of positive operators. New York J. Math. 26 (2020), 129–137
[3] HASSI, S.; SEBESTYÉN, Z.; DE SNOO, H. Lebesgue type decompositions for nonnega-
tive forms. J. Funct. Anal. 257 (2009), no. 12, 3858–3894.
[4] JURY, M. T.; MARTIN R.T.W. Lebesgue decomposition of non-commutative measures,
arXiv: 1910.09965.
[5] KOSAKI, H. Lebesgue decomposition of states on a von Neumann algebra. Amer. J. Math.
107 (1985), no. 3, 697–735.
[6] MOLNÁR, L., Maps on positive operators preserving Lebesgue decompositions. Electron.
J. Linear Algebra 18 (2009), 222–232.
[7] TITKOS, T. Arlinskii’s iteration and its applications. Proc. Edinb. Math. Soc. 62 (2019),
no. 1, 125–133.
[8] VOICULESCU, D-V. Lebesgue decomposition of functionals and unique preduals for com-
mutants modulo normed ideals. Houston J. Math. 43 (2017), no. 4, 1251–1262.
213
Maps preserving absolute continuity of positive operators
Tamás Titkos, titkos@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Zsigmond Tarcsay, György Pál Gehér
Motivated by their measure theoretic analogues, Ando introduced in [1] the notions of absolute
continuity and singularity of positive operators, and proved a Lebesgue-type decomposition
theorem. Since then, similar results have been proved in more general contexts [3, 4, 5, 7, 8],
just to mention a few. Molnár in [6] described the structure of bijective maps on the cone of
positive operators that preserve the Lebesgue decomposition in both directions. It turned out that
the cone is quite rigid in the sense that these maps can be always written in the form A → SAS∗
with a bounded, invertible, linear- or conjugate linear operator S. A natural question arises: is
it possible to weaken the preserver property, and to characterize those bijections that preserve
absolute continuity only? The aim of this talk to answer this question in the affirmative.
References
[1] ANDO, T. Lebesgue-type decomposition of positive operators. Acta Sci. Math. (Szeged)
38 (1976), no. 3-4, 253–260.
[2] GY.P. GEHÉR, ZS. TARCSAY, T. TITKOS, Maps preserving absolute continuity and sin-
gularity of positive operators. New York J. Math. 26 (2020), 129–137
[3] HASSI, S.; SEBESTYÉN, Z.; DE SNOO, H. Lebesgue type decompositions for nonnega-
tive forms. J. Funct. Anal. 257 (2009), no. 12, 3858–3894.
[4] JURY, M. T.; MARTIN R.T.W. Lebesgue decomposition of non-commutative measures,
arXiv: 1910.09965.
[5] KOSAKI, H. Lebesgue decomposition of states on a von Neumann algebra. Amer. J. Math.
107 (1985), no. 3, 697–735.
[6] MOLNÁR, L., Maps on positive operators preserving Lebesgue decompositions. Electron.
J. Linear Algebra 18 (2009), 222–232.
[7] TITKOS, T. Arlinskii’s iteration and its applications. Proc. Edinb. Math. Soc. 62 (2019),
no. 1, 125–133.
[8] VOICULESCU, D-V. Lebesgue decomposition of functionals and unique preduals for com-
mutants modulo normed ideals. Houston J. Math. 43 (2017), no. 4, 1251–1262.
213
