Duality of Paranormed Spaces of Matrices Defining Linear Operators from lp into lq
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Let 1 ≤ p, q < ∞ be fixed, and let R = [rjk] be an infinite scalar matrix such that 1 ≤ rjk < ∞ and supj,k rjk < ∞ . Let B(lp, lq) be the set of all bounded linear operator from lp into lq. For a fixed Banach algebra B with identity, we define a new vector space SR p,q(B) of infinite matrices over B and a paranorm G on SR p,q(B) as follows: letSR p,q(B) = n A : A [R] ∈ B(lp, lq) o and G(A) = kA [R] k 1 Mp,q, where A [R] = [kajkkrjk ] and M = max{1, supj,k rjk}. The exis tance of SR p,q(B) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space
Abstract
Let 1 ≤ p, q < ∞ be fixed, and let R = [rjk] be an infinite scalar matrix such that 1 ≤ rjk < ∞ and supj,k rjk < ∞ . Let B(lp, lq) be the set of all bounded linear operator from lp into lq. For a fixed Banach algebra B with identity, we define a new vector space SR p,q(B) of infinite matrices over B and a paranorm G on SR p,q(B) as follows: letSR p,q(B) = n A : A [R] ∈ B(lp, lq) o and G(A) = kA [R] k 1 Mp,q, where A [R] = [kajkkrjk ] and M = max{1, supj,k rjk}. The exis tance of SR p,q(B) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space
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