The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.

The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.

We introduce and study some operational quantities associated to a space ideal A. These quantities are used to define generalized semi-Fredholm operators associated to A, and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that...

In the present paper, we establish first relation between perturbation of upper Fredholm and strictly singular, then lower semi-Fredholm cosingular linear relations. Most importantly in Theorem 3.4, show that P(F?(X,Y)) coincides with SC(X,Y).We bring to light, relationship essential spectra a multivalued operator its selection

We show the existence of Banach spaces X, Y such that the set of strictly singular operators ᏿(X,Y) (resp., the set of strictly cosingular operators Ꮿ᏿(X,Y)) would be strictly included in F + (X,Y) (resp., F − (X,Y)) for the nonempty class of closed densely defined upper semi-Fredholm operators Φ + (X,Y) (resp., for the nonempty class of closed densely defined lower semi-Fredholm operators Φ − ...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.