The Spectrum of Second Order Quantum Difference Operator

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.

Sequence Spaces and Spectrum of q-Difference Operator of Second Order

“…The construction of sequence spaces by using q-calculus was realized very recently in the literature. The following q-analogs (or (p, q)-analogs) C q = (c q rt ), C(q) = ( crt ), ∇ 2 q = (δ 2;q rt ) and E(p, q) = (e p,q rt ) of the Cesàro matrix, Catalan matrix, difference matrix of the second order and Euler matrix, respectively, can be found in [19][20][21][22]:…”

“…By following this theory of the q-difference operator, quite recently, Yaying et al [21] introduced q-difference sequence spaces of the second order c 0 (∇ 2 q ) = (c 0 ) ∇ 2 q and c(∇ 2 q ) = c ∇ 2 q and obtained the spectral analysis of ∇ 2 q over the space c 0 . One can observe from the above discussion that the widely studied spaces c 0 , c, ∞ and p have been modified by various authors with the use of difference operators, as well as investigating the relations of the aforesaid spaces.…”

“…One can observe from the above discussion that the widely studied spaces c 0 , c, ∞ and p have been modified by various authors with the use of difference operators, as well as investigating the relations of the aforesaid spaces. Most recently, involving q-calculus and the difference operators, the authors of [21] presented a modification of c 0 and c, which were denoted by c 0 (∇ 2 q ) and c(∇ 2 q ). In the next section, we present the generalization of ∞ and p by defining p (∇ 2 q ) and ∞ (∇ 2 q ), which will fill the gap of further research in this direction.…”

“…Motivated by the above studies, in particular [8,21], we construct the domains ( p ) ∇ 2 q and ( ∞ ) ∇ 2 q . Additionally, the spectral analysis of the operator ∇ 2 q over the space 1 is also obtained.…”

“…q ) and ∞ (∇ 2 q ) In this section, the q-difference sequence spaces p (∇ 2 q ) and ∞ (∇ 2 q ) are presented, inclusion relations are obtained, and the basis of the space p (∇ 2 q ) is determined. Yaying et al [21,26] defined the difference operator ∇ 2 q : ω → ω by…”