The algebra of bounded linear operators on ℓp ⊕ ℓq has infinitely many closed ideals

We prove that in the reflexive range 1 < p < q < ∞, the algebra ℒ(ℓp⊕ℓq) of all bounded linear operators on ℓp⊕ℓq has infinitely many closed ideals. This solves a problem raised by A. Pietsch [Operator ideals, Math. Monogr. 16, VEB Deutscher Verlag der Wissenschaften, Berlin 1978, Problem 5.3.3] in his book `Operator ideals'.The first author’s research was supported by NSF grant DMS-1160633. The second author was supported by the 2014 Workshop in Analysis and Probability at Texas A&M University

Renorming spaces with greedy bases

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given \vare>0, so that the basis becomes (1+\vare)-democratic, and hence (2+\vare)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1+\vare)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1<p<\infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space

On stability of metric spaces and Kalton's property QQ

The first named author introduced the notion of upper stability for metric spaces as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property $Q$ does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces.Comment: 14 page