## Banach Journal of Mathematical Analysis

### Algebraically paranormal operators on Banach spaces

Pietro Aiena

#### Abstract

In this paper we show that a bounded linear operator on a Banach space $X$ is polaroid if and only if $p(T)$ is polaroid for some polynomial $p$. Consequently, algebraically paranormal operators defined on Banach spaces are hereditarily polaroid. Weyl type theorems are also established for perturbations $f(T+K)$, where $T$ is algebraically paranormal, $K$ is algebraic and commutes with $T$, and $f$ is an analytic function, defined on an open neighborhood of the spectrum of $T+K$, such that $f$ is nonconstant on each of the components of its domain. These results subsume recent results in this area.

#### Article information

Source
Banach J. Math. Anal., Volume 7, Number 2 (2013), 136 -145 .

Dates
First available in Project Euclid: 20 March 2013

https://projecteuclid.org/euclid.bjma/1363784227

Digital Object Identifier
doi:10.15352/bjma/1363784227

Mathematical Reviews number (MathSciNet)
MR3039943

Zentralblatt MATH identifier
1291.47015

Subjects
Primary: 47A10
Secondary: 47A11 47A53 47A55

#### Citation

Aiena , Pietro. Algebraically paranormal operators on Banach spaces. Banach J. Math. Anal. 7 (2013), no. 2, 136 --145. doi:10.15352/bjma/1363784227. https://projecteuclid.org/euclid.bjma/1363784227

#### References

• P. Aiena, Fredholm and local spectral theory, with application to multipliers, Kluwer Acad. Publishers, 2004.
• P. Aiena, Classes of Operators Satisfying $a$-Weyl's theorem, Studia Math. 169 (2005), 105–122.
• P. Aiena, E. Aponte and E. Bazan, Weyl type theorems for left and right polaroid operators, Integral Equations Operator Theory 66 (2010), no. 1, 1-20.
• P. Aiena and E. Aponte. Polaroid type operators under perturbations, Preprint.
• P. Aiena, M. L. Colasante and M. González, Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002), 2701–2710.
• P. Aiena, M. Chō and M. González, Polaroid type operator under quasi-affinities, J. Math. Anal. Appl. 371 (2010), no. 2, 485-495.
• P. Aiena and J. R. Guillen, Weyl's theorem for perturbations of paranormal operators, Proc. Amer. Math. Soc.35, (2007), 2433–2442.
• P. Aiena, J. Guillen and P. Peña, Property $(w)$ for perturbation of polaroid operators, Linear Algebra Appl. 4284 (2008), 1791–1802.
• P. Aiena and J. E. Sanabria, On left and right poles of the resolvent. Acta Sci. Math. (Szeged), 74 (2008), 669–687.
• M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457–465.
• M. Berkani and J.J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359–376.
• N.N. Chourasia and P.B. Ramanujan, Paranormal operators on Banach spaces, Bull. Austral. Math. Soc. 21 (1980), no. 2, 161–168.
• L.A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 20 (1970), 529–544.
• R.E. Curto and Y.M. Han, Weyl's theorem for algebraically paranormal operators, Integral Equations Operator Theory 47 (2003), 307–314.
• B.P. Duggal, Polaroid operators satisfying Weyl's theorem, Linear Algebra Appl. 414 (2006), no. 1, 271-277.
• B.P. Duggal, Hereditarily polaroid operators, SVEP and Weyl's theorem, J. Math. Anal. Appl. 340 (2008), no. 1, 366–373.
• T. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal ans several related classes Sci. Math. 1 (1998), no. 3, 389–403.
• H. Heuser, Functional Analysis, Wiley, 1982.
• K.B. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), 323–336.
• K.B. Laursen and M.M. Neumann, Introduction to local spectral theory, Clarendon Press, Oxford 2000.
• M. Oudghiri, Weyl's and Browder's theorem for operators satisfying the SVEP, Studia Math. 163 (2004), 85–101.
• C. Schmoeger, On totally paranormal operators, Bull. Aust. Math. Soc. 66, (2002), 425-441.
• A. Uchiyama and K. Tanahashi, Bishop's property $(\beta)$ for paranormal operators Oper. Matrices 3 (2009), no 4, 517-524.