Two remarks about nilpotent operators of order two
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- by Stephan Ramon Garcia, Bob Lutz and Dan Timotin PDF
- Proc. Amer. Math. Soc. 142 (2014), 1749-1756 Request permission
Abstract:
We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.References
- Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, and Warren R. Wogen, Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity, Indiana Univ. Math. J. 59 (2010), no. 2, 595–620. MR 2648079, DOI 10.1512/iumj.2010.59.4097
- Stephan Ramon Garcia, Conjugation and Clark operators, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 67–111. MR 2198373, DOI 10.1090/conm/393/07372
- Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285–1315. MR 2187654, DOI 10.1090/S0002-9947-05-03742-6
- Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913–3931. MR 2302518, DOI 10.1090/S0002-9947-07-04213-4
- Stephan Ramon Garcia and William T. Ross, Recent progress on truncated Toeplitz operators, Blaschke products and their applications, Fields Inst. Commun., vol. 65, Springer, New York, 2013, pp. 275–319. MR 3052299, DOI 10.1007/978-1-4614-5341-3_{1}5
- S. R. Garcia, W. T. Ross, and W. R. Wogen, ${C}^*$-algebras generated by truncated Toeplitz operators, Oper. Theory Adv. Appl. 236, 181–192.
- Stephan Ramon Garcia, Means of unitaries, conjugations, and the Friedrichs operator, J. Math. Anal. Appl. 335 (2007), no. 2, 941–947. MR 2345511, DOI 10.1016/j.jmaa.2007.01.094
- Stephan Ramon Garcia, Aluthge transforms of complex symmetric operators, Integral Equations Operator Theory 60 (2008), no. 3, 357–367. MR 2392831, DOI 10.1007/s00020-008-1564-y
- Stephan Ramon Garcia, Daniel E. Poore, and William T. Ross, Unitary equivalence to a truncated Toeplitz operator: analytic symbols, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1281–1295. MR 2869112, DOI 10.1090/S0002-9939-2011-11060-8
- Stephan Ramon Garcia and Warren R. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065–6077. MR 2661508, DOI 10.1090/S0002-9947-2010-05068-8
- Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
- Donald Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491–526. MR 2363975, DOI 10.7153/oam-01-29
- E. Strouse, D. Timotin, and M. Zarrabi, Unitary equivalence to truncated Toeplitz operators, Indiana Univ. Math. J. 61 (2012), no. 2, 525–538. MR 3043586, DOI 10.1512/iumj.2012.61.4562
- S. R. Treil′, An inverse spectral problem for the modulus of the Hankel operator, and balanced realizations, Algebra i Analiz 2 (1990), no. 2, 158–182 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 2, 353–375. MR 1062268
Additional Information
- Stephan Ramon Garcia
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- MR Author ID: 726101
- Email: Stephan.Garcia@pomona.edu
- Bob Lutz
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
- MR Author ID: 1053423
- Email: boblutz@umich.edu
- Dan Timotin
- Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 014700, Romania
- Email: Dan.Timotin@imar.ro
- Received by editor(s): June 25, 2012
- Published electronically: February 19, 2014
- Additional Notes: The first and second authors were partially supported by National Science Foundation Grant DMS-1001614
- Communicated by: Richard Rochberg
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1749-1756
- MSC (2010): Primary 46Lxx, 47A05, 47B35, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-2014-11944-7
- MathSciNet review: 3168480