论文标题
局部核操作员亚代甲的逆闭合性
Inverse-closedness of the subalgebra of locally nuclear operators
论文作者
论文摘要
令$ x $为Banach空间，而$ t $为有限的线性操作员，以$ L_P（\ Mathbb Z^C，X）$，$ 1 \ le p \ le p \ le \ infty $。如果可以以\ begin {equination*}（tx）_k = \ sum \ limits_ {m \ in \ mathbb z^c} b_} b _} b_m} b_m} b_m} km {km} x_________ {k-m} x { $ b_ {km}：\，x \至x $是核的，\ begin {equination*} \ lvert b_ {km} \ rvert _ {\ Mathfrak s_1} \ Mathfrak S_1} \leβ_{mleβ_} $ \ lvert \ cdot \ rvert _ {\ mathfrak s_1} $是核定标准，$β\ in l_ {1}（\ Mathbb z^c，\ Mathbb z^c，\ Mathbb c）$或$β\ in L_ {1，G}（g}（g}（g}）） z^c $。已经确定，如果$ t $是本地核的，并且操作员$ \ mathbf1+t $是可逆的，则逆操作员$（\ mathbf1+t）^{ - 1} $具有$ \ mathbf1+t_1 $的形式，其中$ t_1 $也是本地核核的。对于运营商在$ l_p（\ mathbb r^c，\ mathbb c）$中作用的情况，该结果将得到完善。
Let $X$ be a Banach space and $T$ be a bounded linear operator acting in $l_p(\mathbb Z^c,X)$, $1\le p\le\infty$. The operator $T$ is called \emph{locally nuclear} if it can be represented in the form \begin{equation*} (Tx)_k=\sum\limits_{m\in\mathbb Z^c} b_{km}x_{k-m},\qquad k\in\mathbb Z^c, \end{equation*} where $b_{km}:\,X\to X$ are nuclear, \begin{equation*} \lVert b_{km}\rVert_{\mathfrak S_1}\leβ_{m},\qquad k,m\in\mathbb Z^c, \end{equation*} $\lVert\cdot\rVert_{\mathfrak S_1}$ is the nuclear norm, $β\in l_{1}(\mathbb Z^c,\mathbb C)$ or $β\in l_{1,g}(\mathbb Z^c,\mathbb C)$, and $g$ is an appropriate weight on $\mathbb Z^c$. It is established that if $T$ is locally nuclear and the operator $\mathbf1+T$ is invertible, then the inverse operator $(\mathbf1+T)^{-1}$ has the form $\mathbf1+T_1$, where $T_1$ is also locally nuclear. This result is refined for the case of operators acting in $L_p(\mathbb R^c,\mathbb C)$.
