In 2012, Găvruţa introduced the notions of K-frame and of atomic system for a linear bounded operator K in a Hilbert space \({\mathcal{H}}\), in order to decompose its range \(\mathcal {R}(K)\) with a frame-like expansion. In this article, we revisit these concepts for an unbounded and densely defined operator \(A:\mathcal {D}(A)\to {\mathcal{H}}\) in two different ways. In one case, we consider a non-Bessel sequence where the coefficient sequence depends continuously on \(f\in \mathcal {D}(A)\) with respect to the norm of \({\mathcal{H}}\). In the other case, we consider a Bessel sequence and the coefficient sequence depends continuously on \(f\in \mathcal {D}(A)\) with respect to the graph norm of A.

中文翻译：

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适用于无限制运算符的框架和弱框架

2012年，Găvruţa引入了Hilbert空间\（{\ mathcal {H}} \）中线性有界算子K的K框架和原子系统的概念，以分解其范围\（\ mathcal {R} （K）\）具有类似框架的展开。在本文中，我们以两种不同的方式重新审视了无界且密集定义的运算符\（A：\ mathcal {D}（A）\ to {\ mathcal {H}} \\）的这些概念。在一种情况下，我们考虑非贝塞尔序列，其中系数序列相对于\（{\ mathcal {H}} \}的范数连续取决于\（f \ in \ mathcal {D}（A）\）。在另一种情况下，我们考虑贝塞尔序列，系数序列连续取决于关于A的图范数\（f \ in \ mathcal {D}（A）\）。