Daniele Puglisi

Department of Mathematics and Computer Sciences
University of Catania
Viale Andrea Doria, 6
I-95125 Catania CT

Office: 346
Office phone: ++39 095 7383055
Fax: ++39 095 330094

E-mail: dpuglisiunict.it

Research

We say that a triple (X, Y, Z) of Banach spaces has the compact operators transitivity property if whenever every operator form X to Y is compact and every operator from Y to Z is compact then every operator form X to Z must be compact. In

• R. Alencar, R. M. Aron, and G. Fricke, Tensor products of Tsirelson’s space, Illinois J. Math. 31 (1987), 17–23.

it is proved that for suitable q and p, and for any Banach space X, the triple (X, lp,lq) has the compact operators transitivity. In

• D. Puglisi; Operators on Bourgain-Delbaen's spaces. Math. Mech. Compl. Sys. 6 (2018), no. 1, 61–67.

it has been proved that, for suitable reals a, b, the triple (l2, Xa,b, l2) as well as (Xa,b, l2, Xa,b) do not have he compact operators transitivity, where Xa,b denotes the Bourgain–Delbaen space.

Questions

• Let Y and Z be reflexive Banach spaces. Does, for every Banach space X, the triple (X, Y, Z) have the compact operators transitivity property?
• Does there exist a pair of Banach spaces (X,Y) such that every n-homogeneous polynomial from X to Y and from Y to X must be compact?

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