## Annals of Functional Analysis

### On class $\mathcal{A}(k^{*})$ operators

#### Abstract

This paper deals with some classes of bounded linear operators on Hilbert spaces. The main emphasis is put onto the classes $\mathcal{A}(k^{*})$ and $\mathcal{A}_{(k^{*})}P,\, k>0$. Some additional results are given for other classes, like $P\mathcal{A}(k^{*})$, $M-\mathcal{A}(k^{*})$ and spectral properties of operators belonging to $\mathcal{A}(k^{*})$ are considered. We also describe under what conditions a matrix-operator $T_{A,B}$ belongs to $\mathcal{A}(k^{*})$, $\mathcal{A}_{(k^{*})}P$ or $P\mathcal{A}(k^{*})$.

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 4 (2015), 90-106.

Dates
First available in Project Euclid: 1 July 2015

https://projecteuclid.org/euclid.afa/1435764004

Digital Object Identifier
doi:10.15352/afa/06-4-90

Mathematical Reviews number (MathSciNet)
MR3365984

Zentralblatt MATH identifier
1320.47022

#### Citation

Braha, Naim L.; Hoxha, Ilmi; Mecheri, Salah. On class $\mathcal{A}(k^{*})$ operators. Ann. Funct. Anal. 6 (2015), no. 4, 90--106. doi:10.15352/afa/06-4-90. https://projecteuclid.org/euclid.afa/1435764004

#### References

• T. Ando, Operators with a norm condition, Acta Sci. Math.(Szeged) 33 (1972), 169–178.
• S.C. Arora and J.K. Thukral, $M^{*}$-Paranormal operators, Glas. Math. Ser. III 22(42), no.1. (1987), 123–129.
• S.C. Arora and J.K. Thukral, On a class of operators, Glas. Math. Ser. III 21(41) (1986), no.2, 381–386.
• S.C. Arora and R. Kumar, $M$-Paranormal operators, Publ. Inst. Math., Nouvelle serie 29 43 (1981), 5–13.
• N. Braha, M. Lohaj, F. Marevci and Sh. Lohaj, Some properties of paranormal and hyponormal operators, Bull. Math. Anal. Appl., V.1, Issue 2 (2009), 23–35.
• J. Diestel and J.J. Uhl, Vector measure, Providence, Rhode Island, 1977.
• L. Debnath and P. Mikusinski, Hilbert spaces with applications, Third edition, Elsevier academic press, 2005.
• B.P. Dugall, I.H. Jeon and I.H. Kim, On $*$-paranormal contractions and properties for $*$-class A operators, Linear Alg. Appl. 436 (2012), 954–962.
• T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594–598.
• 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), no.3, 389–403.
• I. Hoxha and N.L. Braha A note on $k$-quasi-$\ast$-paranormal operators, J. Inequal. Appl. 2013, 2013:350.
• Y.M. Han and A.-H. Kim, A note on $*$-paranormal operators, Integral Equations Operator Theory 49 (2004), no.4, 435–444.
• V. Istratescu, T. Saito and T. Yoshino, On a class of operators, Tohoku Math. J. 18 (1966), 410–413.
• I.H. Kim, Weyl's theorem and tensor product for operators satisfying $T^{*k}|T^{2}|T^{k} \geq T^{*k}|T|^{2}T^{k}$, J. Korean Math. Soc. 47 (2010), No. 2, 351–361.
• I.H. Kim, On spectral continuities and tensor products of operators, Journal of the Chungcheong Mathematical Society, Volume 24,No.1, March 2011.
• K.B. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), 323–336.
• K.B. Laursen and M.M. Neumann, An introduction to Local Spectral Theory, London Mathematical Society Monographs, Oxford 2000.
• T.-W. Ma, Banach$-$Hilbert spaces, Vector measures and Group Representation, World Scientific, 2002.
• C.A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271.
• S. Mecheri and S. Makhlouf, Weyl Type theorems for posinormal operators, Math. Proc. Royal Irish. Acad. 108 (2008),no.1, 68–79.
• S. Mecheri, On quasi-$*$-paranormal operators, Ann. Funct. Anal 3 (2012), 86–91.
• S. Panayappan and A. Radharamani, A Note on $p$-$*$-paranormal Operators and Absolute-$k^{*}$-Paranormal Operators, Int. J. Math. Anal. 2 (2008), no. 25-28, 1257–1261.
• V. Rako\~cevi\`c, On the essential approximate point spectrum II, Math Vesnik 36 (1984), 89–97.
• T. Saito, Hyponormal operators and Related topics, Lecture notes in Mathematics, Springer-Verlag, 247, 1971.
• J. Stochel Seminormality of operators from their tensor products, Proc. Amer. Math. 124 (1996), 435–440.
• K. Tanahashi, On log-hyponormal operators, Integral Equations Operator Theory 34 (1999), 364–372.