Injective isometries in Orlicz spaces

New axiomatizable classes of Banach spaces via disjointness-preserving isometries

Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, …

On universal left-stability of $ε$-isometries

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $γ>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leqγ\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. I…

Weighted composition operators on discrete weighted Banach spaces

We present the current results in the study of weighted composition operators on weighted Banach spaces of an unbounded, locally finite metric space. Specifically, we determine characterizations of bounded and compact weighted composition operators, including the operator and essential norms. In addition, we characterize the weighted composition operators that are injective, are bounded below, have closed range, and have bounded inverse. We characterize the isometries and surjective isometries among the weighted composition operators, as well as those that satisfy the Fredholm condition. Lastl…

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…

Universal stability of Banach spaces for $\varepsilon$-isometries

Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard $\varepsilon$-isometry $f:X\rightarrow Y$ is said to be $(α,γ)$-stable (with respect to $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ for some $α, γ>0$) if $T$ is a linear operator with $\|T\|\leqα$ so that $Tf-Id$ is uniformly bounded by $γ\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(α,γ)$-stable for some $α,γ>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show tha…

量子比特态空间上的量子Wasserstein等距映射

Quantum Wasserstein isometries on the qubit state space

本文研究量子比特态空间上量子Wasserstein距离的等距映射，证明Pauli矩阵成本下的等距群为酉或反酉共轭并对应O(3)，同时发现clock和shift算子成本下存在非满射、非单射的非常规等距映射。