无界线性算子的正则矩阵
Regular matrices of unbounded linear operators
AI总结：
本文针对巴拿赫空间上的线性算子矩阵，建立了Silverman--Toeplitz型定理，揭示了经典正则矩阵刻画与理想收敛下相关变体的关系，还得到矩阵类刻画及Hahn--Schur定理推广。
摘要：
设$X,Y$为巴拿赫空间，固定线性算子$T \in \mathcal{L}(X,Y)$，以及$ω$上的理想$\mathcal{I}, \mathcal{J}$。我们得到关于$\mathcal{L}(X,Y)$中线性算子矩阵$A=(A_{n,k}: n,k \in ω)$的Silverman--Toeplitz型定理，使得对每个$\mathcal{I}$-收敛的$X$-值序列$x=(x_0,x_1,\ldots)$，有$\mathcal{J}\text{-}\lim Ax=T(\hspace{.2mm}\mathcal{I}\text{-}\lim x)$。这建立了经典正则矩阵Silverman--Toeplitz刻画与双序列多维类似物、线性算子矩阵变体及理想收敛标量情形最新版本的关系。作为副产品，得到若干矩阵类的刻画及经典Hahn--Schur定理的推广，证明使用De Bondt和Vernaeve在[J. Math. Anal. Appl. 495 (2021)]中获得的Banach--Steinhaus定理的理想版本。
英文摘要：
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)].
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