Spaceability and operator ideals (original) (raw)

Definition Let B denote the class of all Banach spaces and let L denote the class of all bounded linear operators between Banach spaces. An operator ideal I is a "mapping" I : B × B −→ 2 L satisfying the following conditions: Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 5 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 5 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 5 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Ideal norms Definition An ideal norm defined on an ideal I is a rule • I that assigns to every operator T ∈ I a non-negative number T I satisfying the following conditions: Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 An ideal norm is a norm. Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 6 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 7 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 13 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 13 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 18 / 24 Consequences Corollary Let E and F be Banach spaces, and {I p : p ∈ [a, b]} be a family of operator ideals such that I p (E , F) I q (E , F) if p < q with continuous inclusion. Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 19 / 24 Consequences Corollary Let E and F be Banach spaces, and {I p : p ∈ [a, b]} be a family of operator ideals such that I p (E , F) I q (E , F) if p < q with continuous inclusion. If E or F is a σ-reproducible Banach space and I b (E , F) is complete for an ideal norm, then the set I b (E , F) \ p<b I p (E , F) is spaceable. Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 19 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 22 / 24 Víctor M. Sánchez (U.C.M.) Genericity and small sets in analysis 2015 May 27th 23 / 24