Abstract
In this study, we give another generalization of second order backward difference operator ∇2 by introducing its quantum analog ∇q2. The operator ∇q2 represents the third band infinite matrix. We construct its domains c0(∇q2) and c(∇q2) in the spaces c0 and c of null and convergent sequences, respectively, and establish that the domains c0(∇q2) and c(∇q2) are Banach spaces linearly isomorphic to c0 and c, respectively, and obtain their Schauder bases and α-, β- and γ-duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇q2 over the Banach space c0 of null sequences.
Highlights
PreliminariesThe q-analog of a mathematical expression means the generalization of that expression by using the parameter q
Due to the vast applications in mathematics, physics and engineering sciences of q-calculus, numerous researchers are engaged in the field
Motivated by the above studies, we present quantum difference operator of the second order and study its domain in the spaces c and c0 of convergent and null sequences, respectively, and determine its spectrum, point spectrum, residual spectrum, and continuous spectrum over the space c0
Summary
The q-analog of a mathematical expression means the generalization of that expression by using the parameter q. The generalized expression returns the original expression when q approaches 1−. The study of q-calculus can be traced back to the time of Euler. It is a wide and interesting area of research in recent times. Due to the vast applications in mathematics, physics and engineering sciences of q-calculus, numerous researchers are engaged in the field. In the field of mathematics, it is widely used by researchers in approximation theory, combinatorics, hypergeometric functions, operator theory, special functions, quantum algebras, etc. The q-analog (nk)q of binomial coefficient (nk) is defined by conditions of the Creative Commons. For detailed studies in q-calculus, readers may see [1]. Since the q-calculus and time scale calculus are correlated, readers can see [2–5] and references therein for more information
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