On operators satisfying the generalized Cauchy-Schwarz inequality. (English) Zbl 1380.47014
Summary: 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:
| [1] | Aluthge, Ariyadasa, On \(p\)-hyponormal operators for \(0<p<1\), Integral Equations Operator Theory, 13, 3, 307-315 (1990) · Zbl 0718.47015 · doi:10.1007/BF01199886 |
| [2] | Aluthge, Ariyadasa; Wang, Derming, \(w\)-hyponormal operators, Integral Equations Operator Theory, 36, 1, 1-10 (2000) · Zbl 0938.47021 · doi:10.1007/BF01236285 |
| [3] | Ando, T., Operators with a norm condition, Acta Sci. Math. (Szeged), 33, 169-178 (1972) · Zbl 0244.47021 |
| [4] | Foia{\c{s}}, Ciprian; Jung, Il Bong; Ko, Eungil; Pearcy, Carl, Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math., 209, 2, 249-259 (2003) · Zbl 1066.47037 · doi:10.2140/pjm.2003.209.249 |
| [5] | Furuta, Takayuki, On the class of paranormal operators, Proc. Japan Acad., 43, 594-598 (1967) · Zbl 0163.37706 |
| [6] | Furuta, Takayuki, Invitation to linear operators, x+255 pp. (2001), Taylor & Francis, Ltd., London · Zbl 1029.47001 · doi:10.1201/b16820 |
| [7] | Halmos, Paul Richard, A Hilbert space problem book, Graduate Texts in Mathematics 19, xvii+369 pp. (1982), Springer-Verlag, New York-Berlin · Zbl 0496.47001 |
| [8] | Halmos, Paul R., Introduction to Hilbert space and the theory of spectral multiplicity, 114 pp. (1998), AMS Chelsea Publishing, Providence, RI · Zbl 0962.46013 |
| [9] | Jung, Il Bong; Ko, Eungil; Pearcy, Carl, Aluthge transforms of operators, Integral Equations Operator Theory, 37, 4, 437-448 (2000) · Zbl 0996.47008 · doi:10.1007/BF01192831 |
| [10] | L{\`“o}wner, Karl, \'”Uber monotone Matrixfunktionen, Math. Z., 38, 1, 177-216 (1934) · Zbl 0008.11301 · doi:10.1007/BF01170633 |
| [11] | McCarthy, Charles A., \(c_p\), Israel J. Math., 5, 249-271 (1967) · Zbl 0156.37902 |
| [12] | Tanahashi, K{\^o}tar{\^o}, On log-hyponormal operators, Integral Equations Operator Theory, 34, 3, 364-372 (1999) · Zbl 0935.47015 · doi:10.1007/BF01300584 |
| [13] | Watanabe, Hideharu, Operators characterized by certain Cauchy-Schwarz type inequalities, Publ. Res. Inst. Math. Sci., 30, 2, 249-259 (1994) · Zbl 0815.47028 · doi:10.2977/prims/1195166132 |
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