ABSTRACT

We investigate the operators T on a Hilbert space $\mathcal{H}$ which have 2-isometric liftings S with the property $S^{\ast }S\mathcal{H}\subset \mathcal{H}$. We show that such liftings are closely related to some extensions of T, which have $2\times 2$ upper triangular block matrices having contractive entries. We describe these extensions by the corresponding 2-isometric liftings, and respectively as regular A-contractions for some positive operators A. Also, we show that some extensions have Brownian isometric (or unitary) liftings and, in particular, we refer to expansive operators with such liftings.

中文翻译：

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布朗环境中算子的提升和延伸

摘要

我们研究希尔伯特空间上的算子T$\mathcal{H}$具有 2 个等距提升S的属性${\mathrm{小号}}^{*}\mathrm{小号}\mathcal{H}\subset \mathcal{H}$. 我们表明这种提升与T的一些扩展密切相关，这些扩展具有$2\times 2$具有收缩项的上三角块矩阵。我们通过相应的 2 等距提升来描述这些扩展，并分别将这些扩展描述为一些正算子A 的常规A收缩。此外，我们展示了一些扩展具有布朗等距（或单一）提升，特别是，我们指的是具有这种提升的扩展算子。