论文标题
无界线性操作员的常规矩阵
Regular matrices of unbounded linear operators
论文作者
论文摘要
令$ x，y $为banach空间，然后修复\ Mathcal {l}（x，y）$的线性运算符$ t \ in $ $ \ mathcal {i}，\ Mathcal {j} $ on $ω$。我们在矩阵上获得Silverman- toeplitz type theorems $ a =（a_ {n，k}：n，k \ inω）$ in $ \ mathcal {l}（x，y）$，因此ax = t（\ hspace {.2mm} \ mathcal {i} \ text { - } \ lim x）$ x $ x $ x $ valued序列$ x =（x_0，x_1，x_1，\ ldots）$，是$ \ mathcal {i} $ - covengent [and bundent and bundent and and and and and and and and and and anded]。这使我们能够建立经典的Silverman之间的关系 - 对常规矩阵的表征和其对双序列的多维模拟，其用于线性操作员的矩阵的变体，以及在理想收敛的上下文中的最新版本（对于标量）。作为副产品，我们获得了几个矩阵类别的特征和经典的hahn-schur定理的概括。在证明中，我们将使用Banach-Steinhaus定理的理想版本，该版本是De Bondt和Vernaeve最近在[J.〜Math。〜Anal。〜Appl。〜\ Textbf {495}（2021）中获得的。
Let $X,Y$ be Banach spaces, and fix a linear operator $T \in \mathcal{L}(X,Y)$, and ideals $\mathcal{I}, \mathcal{J}$ on $ω$. We obtain Silverman--Toeplitz type theorems on matrices $A=(A_{n,k}: n,k \in ω)$ of linear operators in $\mathcal{L}(X,Y)$, so that $$ \mathcal{J}\text{-}\lim Ax=T(\hspace{.2mm}\mathcal{I}\text{-}\lim x) $$ for every $X$-valued sequence $x=(x_0,x_1,\ldots)$ which is $\mathcal{I}$-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman--Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn--Schur theorem. In the proofs we will use an ideal version of the Banach--Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J.~Math.~Anal.~Appl.~\textbf{495} (2021)].
