The study of operators satisfying Bishop's property $(\beta)$ is of significant interest and is currently being done by a number of mathematicians around the world. Recently Uchiyama and Tanahashi [Oper. Matrices 4 (2009), 517--524] showed that a paranormal operator has Bishop's property $(\beta)$. In this paper we introduce a new class of operators which we call the class of $k$-quasi-paranormal operators. An operator $T$ is said to be a $k$-quasi-paranormal operator if it satisfies $||T^{k+1}x||^{2}\leq||T^{k+2}x|||T^{k}x||$ for all $x\in H$ where k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi-class $A$ operators. We prove basic properties and give a structure theorem of $k$-quasi-paranormal operators. We also show that Bishop's property $(\beta)$ holds for this class of operators. Finally, we prove that if $E$ is the Riesz idempotent for a nonzero isolated point $\lambda_{0}$ of the spectrum of a $k$-quasi-paranormal operator $T$, then $E$ is self-adjoint if and only if the null space of $T-\lambda_{0},\, \ker(T-\lambda_{0})\subseteq \ker(T^{*}-\overline{\lambda_{0}})$.
Banach J. Math. Anal.
6(1):
147-154
(2012).
DOI: 10.15352/bjma/1337014673
T. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. Math. 1 (1998), 389–403. MR1688255 0936.47009 T. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. Math. 1 (1998), 389–403. MR1688255 0936.47009
T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594–598. MR221302 10.3792/pja/1195521514 euclid.pja/1195521514
T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594–598. MR221302 10.3792/pja/1195521514 euclid.pja/1195521514
M.A. Rosenblum, On the operator equation $BX - XA = Q$, Duke Math. J. 23 (1956), 263–269. MR79235 0073.33003 10.1215/S0012-7094-56-02324-9 euclid.dmj/1077466818
M.A. Rosenblum, On the operator equation $BX - XA = Q$, Duke Math. J. 23 (1956), 263–269. MR79235 0073.33003 10.1215/S0012-7094-56-02324-9 euclid.dmj/1077466818
J. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476. MR173161 0139.31201 10.1090/S0002-9947-1965-0173161-3 J. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476. MR173161 0139.31201 10.1090/S0002-9947-1965-0173161-3
K. Tanahashi, Putnam's Inequality for $\log$-hyponormal operators, Integral Equations Operator Theory 43 (2004), 364–372. MR2029946 1062.47031 10.1007/s00020-999-1172-5 K. Tanahashi, Putnam's Inequality for $\log$-hyponormal operators, Integral Equations Operator Theory 43 (2004), 364–372. MR2029946 1062.47031 10.1007/s00020-999-1172-5
A. Uchiyama and K. Tanahashi, Bishop's property $(\beta)$ for paranormal operators, Oper. Matrices 4 (2009), 517–524. MR2597677 1198.47039 10.7153/oam-03-29 A. Uchiyama and K. Tanahashi, Bishop's property $(\beta)$ for paranormal operators, Oper. Matrices 4 (2009), 517–524. MR2597677 1198.47039 10.7153/oam-03-29
A. Uchiyama, On isolated points of the spectrum of paranomal operators, Integral Equations and Operator Theory. 55(2006), 145–151. MR2226642 1105.47021 10.1007/s00020-005-1386-0 A. Uchiyama, On isolated points of the spectrum of paranomal operators, Integral Equations and Operator Theory. 55(2006), 145–151. MR2226642 1105.47021 10.1007/s00020-005-1386-0