Constructing Banach ideals using upper ℓₚ-estimates

Abstract

Using upper ℓp-estimates for normalized weakly null sequence images, we describe a new family of operator ideals 〖ⱲD〗_(lₚ)^((∞,ξ)) with parameters 1 ≤ p ≤ ∞ and 1 ≤ ξ ≤ ω₁. These classes contain the completely continuous operators, and are distinct for all choices 1 ≤ p ≤ ∞ and, when p≠ 1 for infinitely many 1 ≤ ξ ≤ ω₁. For the case ξ = 1, there exists an ideal norm.〖ǁ .ǁ〗_((ₚ₁)) on the class 〖ⱲD〗_(lₚ)^((∞,ξ)) under which it forms a Banach ideal. We also prove that each space 〖ⱲD〗_(lₚ)^((∞,ω₁))(X, Y) is the intersection of the spaces 〖ⱲD〗_(lₚ)^((∞,ξ)) (X, Y) over all 1 ≤ ξ < ω₁.