On second-order q-difference operators

Abstract

The minimal and maximal operators defined by second-order q-difference operator are discussed in this paper. Spectrum sets of these defined operators have been determined. In addition, two extensions of the minimal operator is also mentioned.

Keywords

• q-hyponormal operators
• q-cohyponormal operator
• second order $q$- difference operator
• minimal and maximal operators spectrum

References

1. Ahmad, B., Alsaedi, A., Ntouyas, S. K., A study of second-order q-difference equations with boundary conditions, Adv. Differ. Equ. 1 (2012), 1–10. https://doi:10.1186/1687-1847-2012-35
2. Ahmad, B., Ntouyas, S. K., Boundary value problems for q-difference inclusions, Abstr. Appl. Abstr. Appl. Anal., Article ID 292860 (2011), 1–15. https://doi:10.1155/2011/292860
3. Allahverdiev, B. P., Tuna, H., Qualitative spectral analysis of singular q- Sturm–Liouville operators, Bull. Malays. Math. Sci. Soc., 43(2) (2020), 1391–1402. https://doi.org/10.1007/s40840-019-00747-3
4. 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
5. Aral, A., A generalization of Szasz Mirakyan operators based on q-integers, Math. Comput. Modelling, 47(9-10) (2008), 1052–1062. https://doi.org/10.1016/j.mcm.2007.06.018
6. Bangerezako, G., An Introduction to q-Difference Equations, UCL, Institut de Mathematiques Pures and Appliquees, Seminaire de Mathematiques, Rapport n.354, Louvain, 2008.
7. Bromwich, T. J., I’A., An Introduction to the Theory of Infinite Series. 1st edn. Macmillan, London, 1908.
8. Euler, L., Introductio in Analysin Infinitorum, vol. 1. Lausanne, Switzerland, Bousquet, 1748 (in Latin).

9. Dobrogowska, A., Odzijewicz, A., Second order q-difference equations solvable by factorization method, J. Comput. Appl. Math., 193(1) (2006), 319–346. https://doi.org/10.1016/j.cam.2005.06.009
10. Phillips, G. M., Interpolation and Approximation by Polynomials, New York, Springer, 2003. https://doi.org/10.1007/b97417
11. Jackson, M., On a q-definite integrals, Quart. J. Pure and Appl. Math., 41 (1910), 193–203.
12. Jackson, F. H., On q-functions and a certain difference operator, Trans. Roy. Soc. Edinb., 46(2) (1909), 253–281. https://doi.org/10.1017/S0080456800002751
13. Naimark, M. A., Linear Differential Operators, Part II, Linear Differential Operators in Hilbert Space, Frederick Ungar, New York, 1968.
14. Stein, E. M., Shakarchi R. Complex Analysis. Princeton, NJ, USA, Princeton Univ. Press, 2003.
15. Yu, C., Wang, J., Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives, Adv. Differ. Equ., (2013), 1–11. https://doi:10.1186/1687-1847-2013-124
16. Ota, S., Some classes of q-deformed operators, J. Operator Theory, 48(1) (2002), 151–186.
17. Sertbaş, M., Saral, C., q-Difference operator and Its q-cohyponormality, Complex Anal. Oper. Theory, 14(8) (2020), 84. https://doi.org/10.1007/s11785-020-01043-w