Quotient algebras of Banach operator ideals related to non-classical approximation properties

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

“…We point out that Sinha–Karn p ‐compact operators were introduced by Sinha and Karn [45] as p ‐compact operators and with the notation $$(K_p,\kappa _p)$$. However, we have decided to adopt the terminology and notation from [47, 49] in order to obtain a clear distinction from the classical ideal of p ‐compact operators independently introduced by Fourie and Swart [16] and Pietsch [37], which are denoted by $$\mathcal {K}_p$$ in this paper (see below).…”

“…By applying a classical result of Maurey on absolutely p ‐summing operators and a characterization of the $$\mathcal {SK}_p$$‐AP for dual spaces due to Delgado et al. [9], it is shown in [49, Proposition 3.6] that $$X^*$$ has the $$\mathcal {SK}_p$$‐AP. Consequently, X has the $$\mathcal {K}_{up}$$‐AP by [22, Theorem 1.1] if $$1<p<2$$ and by [31, Theorem 4.7] if $$p=1$$.…”

“…However, our approach is based on a factorization argument of Reinov [43, Lemma 1.1] in contrast to the approach of Choi and Kim in [5]. For other similar constructions based on [43, Lemma 1.1], we refer to [47, Theorem 3.9] and [49, Proposition 4.3].…”

“…The $$\mathcal {I}$$‐AP has recently been studied in several papers for various Banach operator ideals $$\mathcal {I}$$, see, for example, [9, 25, 27, 28, 30, 35], and references therein. Moreover, results about special instances of the $$\mathcal {I}$$‐AP have been used in the recent studies [47, 49] of radical quotient algebras of Banach operator ideals.…”