Joanna Garbulińska-Wegrzyn

arXiv2019

A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $Ω$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $Ω$ is generic in the sense of a natural infinite game.

math.FA↗

arXiv2014

Quasi-Banach spaces of almost universal disposition

We show that for each $p\in(0,1]$ there exists a separable $p$-Banach space $\mathbb G_p$ of almost universal disposition, that is, having the following extension property: for each $ε>0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of $\mathbb G_p$, there is an $ε$-isometry $f:Y\to \mathbb G_p$ such that $x=f(g(x))$ for all $x\in X$. Such a space is unique, up to isometries, does contain an isometric copy of each separable $p$-Banach space and has the remarkable property of being "locally injective" amongst $p$-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and "separably injective". No separably injective $p$-Banach space was previously known for $p<1$.

math.FA↗