$q$-Difference Operator on $L_q^{2}( 0, + \infty )$

Year 2023, Volume: 72 Issue: 1, 247 - 258, 30.03.2023

Meltem Sertbaş Coşkun Saral

https://doi.org/10.31801/cfsuasmas.1121701

Abstract

In this research, the minimal and maximal operators defined by $q$- difference expression are given in the Hilbert space $L_q^{2}( 0, + \infty )$. The existence problem of a $q^{-1}$-normal extension for the minimal operator is mentioned. In addition, the sets of the minimal operator spectrum and the maximal operator spectrum are examined.

Keywords

$q$- difference operator, minimal operator, maximal operator, $q$-formally normal operator, $q$-normal operator, spectrum

References

• Annaby, M. H, Mansour, Z. S., q-Fractional Calculus and Equations, Lecture Notes in Mathematics, vol. 2056, Springer, Heidelberg 2012. https://doi.org/10.1007/978-3-642-30898-7
• Cimpric, J., Savchuk, Yu., Schmüdgen, K., On q-normal operators and the quantum complex plane,Trans. Amer. Math. Soc., 366(1) (2014), 135–158. https://doi.org/10.1090/S0002-9947-2013-05733-9
• Ernst, T., The History of q-Calculus and a New Method, U.U.D.M. Report 2000, 16, Uppsala, Department of Mathematics, Uppsala University 2000.
• Euler, L., Introduction in Analysin Infinitorum, vol. 1. Lausanne, Switzerland, Bousquet 1748 (in Latin).
• Gauss, C. F., Disquisitiones generales circa seriem infinitam, Werke, (1813), 124-162.
• Hörmander, L., On the theory of general partial differential operators, Acta Math. 94 (1955), 161-248. https://doi.org/10.1007/BF02392492
• Jackson, F. H., On q-functions and a certain difference operator, Trans. Roy. Soc. Edinb., 46 (1908), 64-72.
• Kac V., Cheung P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4613-0071-7
• Lavagno A., Scarfone, A. M., q-deformed structure and generalized thermodynamics, Rep. Math. Phys. 55(3) (2005), 423-433. https://doi.org/10.1016/S0034-4877(05)80056-4
• Ota, S., Some classes of q-deformed operators, J. Operator Theory, 48(1) (2002), 151-186.
• Ota, S., Szafraniec, F. H., Notes on q-deformed operators, Studia Math., 165(3) (2004), 295-301. https://doi.org/10.4064/sm165-3-7
• Ota, S., Szafraniec F. H., q-positive definiteness and related topics, J. Math. Anal. Appl., 329(2) (2007), 987-997. https://doi.org/10.1016/j.jmaa.2006.07.006
• Schmüdgen, K., An Invitation to Unbounded Representations of ∗-Algebras on Hilbert space. Springer Cham, 2020. https://doi.org/10.1007/978-3-030-46366-3
• Sertbaş, M., Yilmaz, F., q-quasinormal operators and its extended eigenvalues, MJM, 2(1) (2020), 9-13.
• Stein, E. M., Shakarchi, R., Complex Analysis, Princeton, NJ, USA, Princeton Univ. Press 2003.