论文标题
欧几里得约旦代数的Korovkin型结果和双随机变化
Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras
论文作者
论文摘要
众所周知的科罗夫金定理断言，如果$ \ {t_k \} $是$ c [a，b] $上的一系列积极的线性转换，以至于$ t_k（h）\ rightarrow h $（在$ c [a，b] $上的sup-norm in \ in \ in \ in \ in \ in $ c [a，b] $） $ [a，b] $，然后$ t_k（h）\ rightArrow h $用于c [a，b] $中的所有$ h \。特别是，如果$ t $是$ c [a，b] $上的正线性转换，使得$ t（h）= h $的所有$ h \ in \ {1，ϕ，ϕ，ϕ^2 \} $，则$ t $是身份转换。在本文中，我们在Euclidean Jordan代数上介绍了这些结果的一些类似物。我们表明，如果$ t $是欧几里得约旦代数$ v $的积极线性转换Jordan框架对应于$ p $的光谱分解；因此，如果正线性转化与约旦框架上的身份变换一致，则它是双重随机的。我们还提出了顺序和弱化的版本。
A well-known theorem of Korovkin asserts that if $\{T_k\}$ is a sequence of positive linear transformations on $C[a,b]$ such that $T_k(h)\rightarrow h$ (in the sup-norm on $C[a,b]$) for all $h\in \{1,ϕ,ϕ^2\}$, where $ϕ(t)=t$ on $[a,b]$, then $T_k(h)\rightarrow h$ for all $h\in C[a,b]$. In particular, if $T$ is a positive linear transformation on $C[a,b]$ such that $T(h)=h$ for all $h\in \{1,ϕ,ϕ^2\}$, then $T$ is the Identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if $T$ is a positive linear transformation on a Euclidean Jordan algebra $V$ such that $T(h)=h$ for all $h\in \{e,p,p^2\}$, where $e$ is the unit element in $V$ and $p$ is an element of $V$ with distinct eigenvalues, then $T=T^*=I$ (the Identity transformation) on the span of the Jordan frame corresponding to the spectral decomposition of $p$; consequently, if a positive linear transformation coincides with the Identity transformation on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.
