On operators satisfying the generalized Cauchy-Schwarz inequality
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- by Hanna Choi, Yoenha Kim and Eungil Ko
- Proc. Amer. Math. Soc. 145 (2017), 3447-3453
- DOI: https://doi.org/10.1090/proc/13473
- Published electronically: January 31, 2017
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Abstract:
In this paper, we introduce the generalized Cauchy-Schwarz inequality for an operator $T\in {\mathcal {L(H)}}$ and investigate various properties of operators which satisfy the generalized Cauchy-Schwarz inequality. In particular, every $p$-hyponormal operator satisfies this inequality. We also prove that if $T\in {\mathcal {L(H)}}$ satisfies the generalized Cauchy-Schwarz inequality, then $T$ is paranormal. As an application, we show that if both $T$ and $T^{\ast }$ in ${\mathcal {L(H)}}$ satisfy the generalized Cauchy-Schwarz inequality, then $T$ is normal.References
- Ariyadasa Aluthge, On $p$-hyponormal operators for $0<p<1$, Integral Equations Operator Theory 13 (1990), no. 3, 307–315. MR 1047771, DOI 10.1007/BF01199886
- Ariyadasa Aluthge and Derming Wang, $w$-hyponormal operators, Integral Equations Operator Theory 36 (2000), no. 1, 1–10. MR 1736916, DOI 10.1007/BF01236285
- T. Ando, Operators with a norm condition, Acta Sci. Math. (Szeged) 33 (1972), 169–178. MR 320800
- Ciprian Foiaş, Il Bong Jung, Eungil Ko, and Carl Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math. 209 (2003), no. 2, 249–259. MR 1978370, DOI 10.2140/pjm.2003.209.249
- Takayuki Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594–598. MR 221302
- Takayuki Furuta, Invitation to linear operators, Taylor & Francis Group, London, 2001. From matrices to bounded linear operators on a Hilbert space. MR 1978629, DOI 10.1201/b16820
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952, DOI 10.1007/978-1-4684-9330-6
- Paul R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1957) edition. MR 1653399
- Il Bong Jung, Eungil Ko, and Carl Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory 37 (2000), no. 4, 437–448. MR 1780122, DOI 10.1007/BF01192831
- Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
- Charles A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271. MR 225140, DOI 10.1007/BF02771613
- Kôtarô Tanahashi, On log-hyponormal operators, Integral Equations Operator Theory 34 (1999), no. 3, 364–372. MR 1689394, DOI 10.1007/BF01300584
- Hideharu Watanabe, Operators characterized by certain Cauchy-Schwarz type inequalities, Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, 249–259. MR 1265473, DOI 10.2977/prims/1195166132
Bibliographic Information
- Hanna Choi
- Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 03760 Korea
- Email: rms5835@gmail.com
- Yoenha Kim
- Affiliation: Institute of Mathematical Sciences, Ewha Womans University, Seoul, 03760 Korea
- MR Author ID: 800848
- Email: yoenha@ewhain.net
- Eungil Ko
- Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 03760 Korea
- MR Author ID: 353576
- Email: eiko@ewha.ac.kr
- Received by editor(s): April 27, 2016
- Received by editor(s) in revised form: September 9, 2016
- Published electronically: January 31, 2017
- Additional Notes: This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT $\&$ Future Planning (2015R1C1A1A02036456).
- Communicated by: Stephan Ramon Garcia
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3447-3453
- MSC (2010): Primary 47A63; Secondary 47B20
- DOI: https://doi.org/10.1090/proc/13473
- MathSciNet review: 3652797