Diagonals of operators and Blaschke’s enigma
HTML articles powered by AMS MathViewer
- by Vladimir Müller and Yuri Tomilov PDF
- Trans. Amer. Math. Soc. 372 (2019), 3565-3595 Request permission
Abstract:
We introduce new techniques allowing one to construct diagonals of bounded Hilbert space operators and operator tuples under “Blaschke-type” assumptions. This provides a new framework for a number of results in the literature and identifies (often large) subsets in the set of diagonals of arbitrary bounded operators (and their tuples). Moreover, our approach leads to substantial generalizations of the results due to Bourin, Herrero, and Stout having assumptions of a similar nature.References
- Joel Hilary Anderson, DERIVATIONS, COMMUTATORS AND THE ESSENTIAL NUMERICAL RANGE, ProQuest LLC, Ann Arbor, MI, 1971. Thesis (Ph.D.)–Indiana University. MR 2620864
- J. H. Anderson and J. G. Stampfli, Commutators and compressions, Israel J. Math. 10 (1971), 433–441. MR 312312, DOI 10.1007/BF02771730
- J. Antezana, P. Massey, M. Ruiz, and D. Stojanoff, The Schur-Horn theorem for operators and frames with prescribed norms and frame operator, Illinois J. Math. 51 (2007), no. 2, 537–560. MR 2342673
- Martín Argerami and Pedro Massey, Schur-Horn theorems in $\textrm {II}_\infty$-factors, Pacific J. Math. 261 (2013), no. 2, 283–310. MR 3037568, DOI 10.2140/pjm.2013.261.283
- Martín Argerami, Majorisation and the Carpenter’s theorem, Integral Equations Operator Theory 82 (2015), no. 1, 33–49. MR 3335507, DOI 10.1007/s00020-014-2180-7
- William Arveson, Diagonals of normal operators with finite spectrum, Proc. Natl. Acad. Sci. USA 104 (2007), no. 4, 1152–1158. MR 2303566, DOI 10.1073/pnas.0605367104
- William Arveson and Richard V. Kadison, Diagonals of self-adjoint operators, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 247–263. MR 2277215, DOI 10.1090/conm/414/07814
- Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, The structured singular value for linear input/output operators, SIAM J. Control Optim. 34 (1996), no. 4, 1392–1404. MR 1395840, DOI 10.1137/S0363012994268825
- H. Bercovici, C. Foias, and C. Pearcy, Dilation theory and systems of simultaneous equations in the predual of an operator algebra. I, Michigan Math. J. 30 (1983), no. 3, 335–354. MR 725785, DOI 10.1307/mmj/1029002909
- B. V. Rajarama Bhat and Mohan Ravichandran, The Schur-Horn theorem for operators with finite spectrum, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3441–3453. MR 3238420, DOI 10.1090/S0002-9939-2014-12114-9
- Jean-Christophe Bourin, Compressions and pinchings, J. Operator Theory 50 (2003), no. 2, 211–220. MR 2050126
- Jean-Christophe Bourin and Eun-Young Lee, Pinchings and positive linear maps, J. Funct. Anal. 270 (2016), no. 1, 359–374. MR 3419765, DOI 10.1016/j.jfa.2015.06.025
- Marcin Bownik and John Jasper, Characterization of sequences of frame norms, J. Reine Angew. Math. 654 (2011), 219–244. MR 2795756, DOI 10.1515/CRELLE.2011.035
- Marcin Bownik and John Jasper, Constructive proof of the Carpenter’s theorem, Canad. Math. Bull. 57 (2014), no. 3, 463–476. MR 3239108, DOI 10.4153/CMB-2013-037-x
- Marcin Bownik and John Jasper, Existence of frames with prescribed norms and frame operator, Excursions in harmonic analysis. Vol. 4, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2015, pp. 103–117. MR 3411093
- Marcin Bownik and John Jasper, The Schur-Horn theorem for operators with finite spectrum, Trans. Amer. Math. Soc. 367 (2015), no. 7, 5099–5140. MR 3335412, DOI 10.1090/S0002-9947-2015-06317-X
- John B. Conway, A course in operator theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, Providence, RI, 2000. MR 1721402, DOI 10.1090/gsm/021
- Catalin Dragan and Victor Kaftal, Sums of equivalent sequences of positive operators in von Neumann factors, Houston J. Math. 42 (2016), no. 3, 991–1017. MR 3570721
- Peng Fan, On the diagonal of an operator, Trans. Amer. Math. Soc. 283 (1984), no. 1, 239–251. MR 735419, DOI 10.1090/S0002-9947-1984-0735419-8
- Peng Fan and Che Kao Fong, An intrinsic characterization for zero-diagonal operators, Proc. Amer. Math. Soc. 121 (1994), no. 3, 803–805. MR 1185279, DOI 10.1090/S0002-9939-1994-1185279-0
- Peng Fan, Che Kao Fong, and Domingo A. Herrero, On zero-diagonal operators and traces, Proc. Amer. Math. Soc. 99 (1987), no. 3, 445–451. MR 875378, DOI 10.1090/S0002-9939-1987-0875378-9
- P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179–192. MR 322534
- C. K. Fong, Diagonals of nilpotent operators, Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 2, 221–224. MR 847875, DOI 10.1017/S0013091500017594
- C. K. Fong, Heydar Radjavi, and Peter Rosenthal, Norms for matrices and operators, J. Operator Theory 18 (1987), no. 1, 99–113. MR 912814
- Domingo A. Herrero, The diagonal entries of a Hilbert space operator, Rocky Mountain J. Math. 21 (1991), no. 2, 857–864. Current directions in nonlinear partial differential equations (Provo, UT, 1987). MR 1121547, DOI 10.1216/rmjm/1181072973
- John Jasper, The Schur-Horn theorem for operators with three point spectrum, J. Funct. Anal. 265 (2013), no. 8, 1494–1521. MR 3079227, DOI 10.1016/j.jfa.2013.06.024
- John Jasper, Jireh Loreaux, and Gary Weiss, Thompson’s theorem for compact operators and diagonals of unitary operators, Indiana Univ. Math. J. 67 (2018), no. 1, 1–27. MR 3776013, DOI 10.1512/iumj.2018.67.6291
- Richard V. Kadison, The Pythagorean theorem. I. The finite case, Proc. Natl. Acad. Sci. USA 99 (2002), no. 7, 4178–4184. MR 1895747, DOI 10.1073/pnas.032677199
- Richard V. Kadison, The Pythagorean theorem. II. The infinite discrete case, Proc. Natl. Acad. Sci. USA 99 (2002), no. 8, 5217–5222. MR 1896498, DOI 10.1073/pnas.032677299
- Victor Kaftal and Gary Weiss, An infinite dimensional Schur-Horn theorem and majorization theory, J. Funct. Anal. 259 (2010), no. 12, 3115–3162. MR 2727642, DOI 10.1016/j.jfa.2010.08.018
- Victor Kaftal and Jireh Loreaux, Kadison’s Pythagorean theorem and essential codimension, Integral Equations Operator Theory 87 (2017), no. 4, 565–580. MR 3648089, DOI 10.1007/s00020-017-2365-y
- Matthew Kennedy and Paul Skoufranis, Thompson’s theorem for $\rm II_1$ factors, Trans. Amer. Math. Soc. 369 (2017), no. 2, 1495–1511. MR 3572280, DOI 10.1090/tran/6711
- M. Kennedy and P. Skoufranis, The Schur-Horn problem for normal operators, Proc. Lond. Math. Soc. (3) 111 (2015), no. 2, 354–380. MR 3384515, DOI 10.1112/plms/pdv030
- Chi-Kwong Li and Yiu-Tung Poon, The joint essential numerical range of operators: convexity and related results, Studia Math. 194 (2009), no. 1, 91–104. MR 2520042, DOI 10.4064/sm194-1-6
- Chi-Kwong Li, Yiu-Tung Poon, and Nung-Sing Sze, Elliptical range theorems for generalized numerical ranges of quadratic operators, Rocky Mountain J. Math. 41 (2011), no. 3, 813–832. MR 2824881, DOI 10.1216/RMJ-2011-41-3-813
- Chi-Kwong Li and Yiu-Tung Poon, Generalized numerical ranges and quantum error correction, J. Operator Theory 66 (2011), no. 2, 335–351. MR 2844468
- Jireh Loreaux and Gary Weiss, Majorization and a Schur-Horn theorem for positive compact operators, the nonzero kernel case, J. Funct. Anal. 268 (2015), no. 3, 703–731. MR 3292352, DOI 10.1016/j.jfa.2014.10.020
- Jireh Loreaux and Gary Weiss, Diagonality and idempotents with applications to problems in operator theory and frame theory, J. Operator Theory 75 (2016), no. 1, 91–118. MR 3474098, DOI 10.7900/jot.2014nov05.2054
- J. Loreaux, Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson, arXiv:1712.06554 (2017).
- Jireh Loreaux, Diagonals of operators: Majorization, a Schur-Horn theorem and zero-diagonal idempotents, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of Cincinnati. MR 3611278
- P. Massey and M. Ravichandran, Multivariable Schur-Horn theorems, Proc. Lond. Math. Soc. (3) 112 (2016), no. 1, 206–234. MR 3458150, DOI 10.1112/plms/pdv067
- Étienne Matheron, Le problème de Kadison-Singer, Ann. Math. Blaise Pascal 22 (2015), no. S2, 151–265 (French). MR 3453286
- Vladimír Müller, The joint essential numerical range, compact perturbations, and the Olsen problem, Studia Math. 197 (2010), no. 3, 275–290. MR 2607493, DOI 10.4064/sm197-3-5
- Vladimir Müller and Yuri Tomilov, Circles in the spectrum and the geometry of orbits: a numerical ranges approach, J. Funct. Anal. 274 (2018), no. 2, 433–460. MR 3724145, DOI 10.1016/j.jfa.2017.10.015
- V. Müller, Yu. Tomilov, Joint numerical ranges and compressions of powers of operators, J. London Math. Soc. 99 (2019), no. 1, 127–152, DOI 10.1112/jlms.12165.
- Vladimir Müller, Spectral theory of linear operators and spectral systems in Banach algebras, 2nd ed., Operator Theory: Advances and Applications, vol. 139, Birkhäuser Verlag, Basel, 2007. MR 2355630
- Andreas Neumann, An infinite-dimensional version of the Schur-Horn convexity theorem, J. Funct. Anal. 161 (1999), no. 2, 418–451. MR 1674643, DOI 10.1006/jfan.1998.3348
- Quentin F. Stout, Schur products of operators and the essential numerical range, Trans. Amer. Math. Soc. 264 (1981), no. 1, 39–47. MR 597865, DOI 10.1090/S0002-9947-1981-0597865-2
- Tin-Yau Tam, A Lie-theoretic approach to Thompson’s theorems on singular values–diagonal elements and some related results, J. London Math. Soc. (2) 60 (1999), no. 2, 431–448. MR 1724865, DOI 10.1112/S0024610799007954
- Shu-Hsien Tso and Pei Yuan Wu, Matricial ranges of quadratic operators, Rocky Mountain J. Math. 29 (1999), no. 3, 1139–1152. MR 1733085, DOI 10.1216/rmjm/1181071625
- Gary Weiss, A brief survey on: 1. infinite-dimensional Schur-Horn theorems and infinite majorization theory with applications to operator ideals; 2. $B(H)$-subideals of operators, Algebraic methods in functional analysis, Oper. Theory Adv. Appl., vol. 233, Birkhäuser/Springer, Basel, 2014, pp. 281–294. MR 3203991, DOI 10.1007/978-3-0348-0502-5_{1}3
Additional Information
- Vladimir Müller
- Affiliation: Institute of Mathematics, Czech Academy of Sciences, Žitna Street 25, Prague, Czech Republic
- Email: muller@math.cas.cz
- Yuri Tomilov
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich Street 8, 00-656 Warsaw, Poland
- MR Author ID: 337361
- Email: ytomilov@impan.pl
- Received by editor(s): May 21, 2018
- Received by editor(s) in revised form: January 8, 2019
- Published electronically: June 3, 2019
- Additional Notes: This work was partially supported by the NCN grant UMO-2017/27/B/ST1/00078 and by Grant No. 17-27844S of GA CR and RVO:67985840
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3565-3595
- MSC (2010): Primary 47A20, 47A12, 47A10; Secondary 47B37
- DOI: https://doi.org/10.1090/tran/7804
- MathSciNet review: 3988619