$q$-Quasinormal Operators and Its Extended Eigenvalues

Year 2020, Volume: 2 Issue: 1, 9 - 13, 30.04.2020

Meltem Sertbaş , Fatih Yılmaz

Abstract

In this paper, the relation between q-deformed quasinormal operators and q-quasinormal operator classes is investigated. Moreover, we proof that these are same. Also, we consider the extended eigenvalue problems for bounded $q$-quasinormal operators.

Keywords

$q$-Quasinormal operator, $q$-Deformed operators, Extended eigenvalue

References

• [1] S. Ota, Some classes of q-deformed operators. J. Operator Theory 48 (2002), 151-186.
• [2] S. Ota and F.K. Szafraniec , Notes on q-deformed operators. Studia Mathematica 165 (3) (2004) , 295-301.
• [3] S. Ota and F.K. Szafraniec, q-Positive definiteness and related operators. j. Math. Anal. Appl. 329 (2007), 987-997.
• [4] S. Ota, On q-deformed hyponormal operators. Math. Nachr. 248-249 (2003), 144-150.
• [5] J. Cimpric, Y. Savchuk and K. Schmudgen, On q-normal operators and quantum complex plane. Trans. Amer. Math. Soc. 366 (2014), 135-158.
• [6] S. Lohaj, Quasi-normal operators. Int. Journal of Math. 4 (47) (2010), 2311-2320.
• [7] J.B. Conway, The theory of subnormal operators. vol. 36. Providence, Rhode Island, USA, American Mathematical Society (1985).
• [8] A. Biswas, A. Lambert and S. Petrovic, Extended eigenvalues and Volterra operators. Glasgn Math. J. 44 (2002), 521-534.
• [9] A. Biswas and S. Petrovic, On extended eigenvalues of operators. Integr. Equat. Oper. Th. 57 (2007), 593-598.
• [10] G. Cassier and H. Alkanjo, Extended spectrum and extended eigenspaces of quasi-normal operators. Banach J. Math. Anal. 11 (2) (2017), 266-281.
• [11] M. Sertbas and F. Yılmaz, On the extended spectrum of some quasinormal operators. Turk. J. Math. 41 (2017), 1477-1481.
• [12] M.T. Karaev, On extended eigenvalues and extended eigenvectors of some operator classes. Proc. Amer. Math. Soc. 134 (8) (2006), 2383-2392.
• [13] M. Gurdal, Description of extended eigenvalues and extended eigenvectors of integration operator on the Wiener algebra. Expo. Math. 27 (2009), 153-160.
• [14] M. Gurdal, On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra. Appl. Math. Lett. 22 (11) (2009), 1727-1729.
• [15] M. Gurdal, Connections between Deddens algebras and extended eigenvectors. Math. Notes. 90 (1) (2011), 37-40.
• [16] M. Karaev, M. G¨urdal and S. Saltan, Some applications of Banach algebra techniques. Math. Nachr. 284 (13) (2011), 1678-1689.
• [17] V.L. Ostrowski and Y.S. Samoilenko, Unbounded operators satisfying non-Lie commutation relations. Rep. Math. Phys. 28 (1989), 91-104.
• [18] V.L. Ostrowski and Y.S. Samoilenko, Representations of quadratic ∗-algebras by bounded and unbounded operators. Rep. Math. Phys. 35 (1995), 283-301 .