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

Abstract

The sequence spaces ℓp(∇q2)(0≤p<∞) and ℓ∞(∇q2) are introduced by using the q-difference operator ∇q2 of the second order. Apart from studying some basic properties of these spaces, we construct the basis and obtain the α-, β- and γ-duals of these spaces. Besides some matrix classes involving q-difference sequence spaces, ℓp(∇q2) and ℓ∞(∇q2) are characterized. The final section is devoted to classifying the spectrum of the q-difference operator ∇q2 over the space ℓ1 of absolutely summable sequences.

On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

“…Then, for all y ∈ p (P (q)), P (q)y exists and is contained in the space ∞ . This implies that G n ∈ p (P (q)) β , (n ∈ N 0 ), which proves the necessity of the relation (11). Now, consider the following equality for all n ∈ N 0 , obtained by using the relation (2)…”

“…Quite recently, some topological and geometric properties of the spaces ( p ) C(q) and ( ∞ ) C(q) have been studied by Yılmaz and Akdemir [10]. Alotaibi et al [11] engineered the sequence spaces p (∇ 2 q ) := ( p ) ∇ 2 q and ∞ (∇ 2 q ) := ( ∞ ) ∇ 2 q of the operator ∇ 2 q in p and ∞ , respectively. We recall that a sequence space X exhibits symmetry (as defined in [12]) if y π(n) belongs to X for any (y n ) in X, where π(n) represents a permutation on N 0 .…”

“…We recall that a sequence space X exhibits symmetry (as defined in [12]) if y π(n) belongs to X for any (y n ) in X, where π(n) represents a permutation on N 0 . Alotaibi et al [11] proved that the space ∞ (∇ 2 q ) does not exhibit symmetricity. The theory of q-sequence spaces is further strengthened by the introduction of q-Catalan sequence space X(C(q)) = X C(q) by Yaying et al [13] for X ∈ {c, c 0 }, and C(q) = cq nv n,v∈N 0 is defined by…”

“…Conversely, assume that the given conditions hold true and choose any y = (y v ) ∈ p (P (q)). The condition (11) means that the sequence G n ∈ p (P (q)) β , which implies that Gy exists. We recall that the space p (P (q)) is linearly isomorphic to the space p .…”