The class of complex symmetric operators is not norm closed
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- by Sen Zhu, Chun Guang Li and You Qing Ji PDF
- Proc. Amer. Math. Soc. 140 (2012), 1705-1708 Request permission
Abstract:
An operator $T\in \mathcal {B(H)}$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal {H}\longrightarrow \mathcal {H}$ so that $CTC=T^*$. In this paper, a class of complex symmetric operators on finite dimensional Hilbert spaces is constructed. As an application, it is shown that Kakutani’s unilateral weighted shift operator is not complex symmetric; however, it is a norm limit of complex symmetric operators. This gives a negative answer to a question of S. Garcia and W. Wogen: that is, whether or not the class of complex symmetric operators is norm closed.References
- Levon Balayan and Stephan Ramon Garcia, Unitary equivalence to a complex symmetric matrix: geometric criteria, Oper. Matrices 4 (2010), no. 1, 53–76. MR 2655004, DOI 10.7153/oam-04-02
- 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, 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 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 R. Garcia and Mihai Putinar, Interpolation and complex symmetry, Tohoku Math. J. (2) 60 (2008), no. 3, 423–440. MR 2453732, DOI 10.2748/tmj/1223057737
- Stephan Ramon Garcia and Warren R. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257 (2009), no. 4, 1251–1260. MR 2535469, DOI 10.1016/j.jfa.2009.04.005
- 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
- T. M. Gilbreath and Warren R. Wogen, Remarks on the structure of complex symmetric operators, Integral Equations Operator Theory 59 (2007), no. 4, 585–590. MR 2370050, DOI 10.1007/s00020-007-1528-7
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Additional Information
- Sen Zhu
- Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
- Address at time of publication: School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, People’s Republic of China
- Email: zhusen@jlu.edu.cn
- Chun Guang Li
- Affiliation: Institute of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
- Email: licg09@mails.jlu.edu.cn
- You Qing Ji
- Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
- Email: jiyq@jlu.edu.cn
- Received by editor(s): January 20, 2011
- Published electronically: September 15, 2011
- Additional Notes: This work was supported by NNSF of China (11026038, 10971079, 11101177) and the Basic Research Foundation of Jilin University (201001001, 201103194).
- Communicated by: Marius Junge
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1705-1708
- MSC (2010): Primary 47A05; Secondary 47B99
- DOI: https://doi.org/10.1090/S0002-9939-2011-11345-5
- MathSciNet review: 2869154