Superstrictly Singular and Superstrictly Cosingular Operators**Partially supported by DAAD foundation

On a problem of Eidelheit from The Scottish Book concerning absolutely continuous functions

“…Obviously, each compact operator is SSS, each SSS operator is strictly singular and T is SSS if it is SSS on a finite codimensional closed subspace (cf. [18]). Then the natural embedding I : L ∞ ֒→ E is SSS.…”

Domination problems for strictly singular operators and other related classes

“…The inclusions L ∞ [0, 1] → E[0, 1] for r.i. spaces E = L ∞ are (non-compact) super strictly singular operators. We refer to Plichko [45] for properties of this operator class.…”

“…Since i is super strictly singular (cf. [45]), the operator T itself is super strictly singular. On the other hand the operator R is not super strictly singular since R n is an isometry for every n. Standard facts show that R is regular and that the inequalities 0 ≤ R − , R + ≤ |R| ≤ T hold true.…”

“…Finally notice that a "super" version of strictly co-singular operators have been also considered in [45], for which domination results can be also deduced (see [21]). …”

Finitely strictly singular operators between James spaces

“…Actually, each property defines an operator ideal. We refer the reader to [1,7,[9][10][11]13] for more information about strictly and finitely strictly singular operators. All the Banach spaces in this paper are assumed to be over real scalars.…”