On (p,r)(p,r)-null sequences and their relatives

Let $1\leq p < \infty$ and $1\leq r \leq p^\ast$, where $p^\ast$ is the conjugate index of $p$. We prove an omnibus theorem, which provides numerous equivalences for a sequence $(x_n)$ in a Banach space $X$ to be a $(p,r)$-null sequence. One of them is that $(x_n)$ is $(p,r)$-null if and only if $(x_n)$ is null and relatively $(p,r)$-compact. This equivalence is known in the "limit" case when $r=p^\ast$, the case of the $p$-null sequence and $p$-compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of $(p,r)$-null sequences

A description of relatively (p; r)-compact sets

We introduce the notion of (p; r)-null sequences in a Banach space and we prove a Grothendieck-like result: a subset of a Banach space is relatively (p; r)-compact if and only if it is contained in the closed convex hull of a (p; r)-null sequence. This extends a recent description of relatively p-compact sets due to Delgado and Piñeiro, providing to it an alternative straightforward proof

Funktsionaalanalüüs

Kopeerimine ja printimine lubatudhttp://www.ester.ee/record=b1194748*es

Quantitative versions of almost squareness and diameter 2 properties

We introduce a quantitative version (using s ∈ 2 (0; 1]) of almost (local) squareness of Banach spaces. The latter concept (i.e., the s = 1 case) was introduced by Abrahamsen, Langemets, and Lima in 2016. Related diameter 2 properties (local, strong, and symmetric strong) are also relaxed correspondingly. Our note contains some (counter-)examples and results for the s-almost (local) squareness property

3-7-aastaste laste vanemate suhtumine ekraanimeedia vahendite kasutamisse koolieelse lasteasutuse õppetöös Põlvamaal

http://www.ester.ee/record=b5150303*es

On M-Ideals of Compact Operators in Lorentz Sequence Spaces

AbstractLet X=d(v,p) and Y=d(w,q) be Lorentz sequence spaces. We investigate when the space K(X,Y) of compact linear operators acting from X to Y forms or does not form an M-ideal (in the space of bounded linear operators). We show that K(X,Y) fails to be a non-trivial M-ideal whenever p=1 or p>q. In the case when 1>p≤q, we establish a general (essential) condition guaranteeing that K(X,Y) is not an M-ideal. In contrast, we prove that non-trivial M-ideals K(X,Y) do exist whenever 1<p<q, and we give a description of them