Closed ideals in the algebra of compact-by-approximable operators

On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators

“…The reverse monotonicity behaviour of the classes K inj p and K up in Proposition 2.3 does not extend to the case of 1 ≤ p < 2 < q < ∞. In fact, in that case there are, according to [44,Proposition 3.6], closed subspaces X ⊂ ℓ p and Y ⊂ ℓ q for which K inj p (X ⊕ Y ) and K inj q (X ⊕ Y ) are incomparable classes of operators. Consequently, also K up (X * ⊕ Y * ) and K uq (X * ⊕ Y * ) are incomparable classes by (2.5) and the first identity in Fact 2.2.(iv).…”

“…We point out that Sinha-Karn p-compact operators were introduced by Sinha and Karn [42] as p-compact operators and with the notation (K p , κ p ) for the corresponding Banach operator ideal. However, we have decided to adopt the terminology and notation from [44,45] in order to obtain a clear distinction from the historically earlier class of (classical) p-compact operators [15,35], which are denoted by K p in this paper (see below).…”

“…Consequently, also K up (X * ⊕ Y * ) and K uq (X * ⊕ Y * ) are incomparable classes by (2.5) and the first identity in Fact 2.2.(iv). Note that the injective hull K inj p is denoted by QK p in [44], see the discussion after [44,Remarks 3.2].…”

“…Following the terminology in [44], we say that the Banach space X has the uniform I-approximation property (uniform I-AP) if…”

“…[9,32,27,24,25,26] and their references. Moreover, results about special instances of the I-AP have been used in the recent studies [44,45] of radical quotient algebras of Banach operator ideals.…”