On models for linear operators
On models for linear operators
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DOI：
10.1002/cpa.3160130309
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发表时间：
1960-08
影响因子：
3
通讯作者：
G. Rota
中科院分区：
文献类型：
--
作者：
G. Rota
S 1. 作者的注释 [3] 中介绍了希尔伯特空间中线性算子模型的概念。在这篇文章中，我们将结合一些极其简单的嵌入技术，给出一些可以从这个概念中获得的显着应用。与[3]相反，这里假设希尔伯特空间是不可分的。我们记得，如果可以找到 H 中具有有界逆的有界线性算子 6'，使得 T= J7-l T, V，则希尔伯特空间 H 中的有界线性算子 T 被称为类似于第二线性算子 TI。当 IlTIl 5 I 时，有界线性算子 T 被称为收缩算子。如果 Tx 在 M 中，则希尔伯特空间 H 的闭子空间 hl 被称为 M 中有界线性算子 T 的不变子空间对于 M 中的每个元素 z。如果 S 是 H 中的第二个有界线性算子，则当存在 (a) S 的不变子空间 M 和 (b) H 中的有界一对一线性算子 I 时，我们说 S 是 H 中线性算子 T 的模型，其范围是闭子空间 M，使得 T 与算子 IG1 一致。坐。 “模型”概念的几个特例最近受到关注。 2 直观上，当 S 对其某个不变子空间的限制给出类似于 T 的算子时，算子 S 就是 T 的模型。在关系“S 是 T 的模型”的明显属性中，我们将使用以下内容：如果 S 是 T 的模型，并且有界线性算子 R 类似于 T，则 S 是 R 的模型。
S 1. The notion of a model for a linear operator in Hilbert space has been introduced in the author’s note [3]. In this note we shall give some remarkable applications which can be obtained from this notion, combined with some exceedingly simple imbedding techniques. Contrary to [3], Hilbert space is not assumed to be separabIe here. We recall that a bounded linear operator T in Hilbert space H is said to be similar to a second linear operator TI if one can find a bounded linear operator 6’with a bounded inverse in H such that T= J7-l T, V. A bounded linear operator T is said to be a contraction operator when IlTIl 5 I. A closed subspace hl of Hilbert space H is said to be an invariant subspace of a bounded linear operator T in M if Tx is in M for every element z in M. If S is a second bounded linear operator in H, we say that S is a model for fhe linear operator T in H when there exist (a) an invariant subspace M of S and (b) a bounded one-to-one linear operator I, in H whose range is the closed subspace M, such that T coincides with the operator IG1. SIT. Several special cases of the notion of “model” have recently received attention. 2 Intuitively, an operator S is a model for T when the restriction of S to some one of its invariant subspaces gives an operator similar to T. Among the obvious properties of the relation “S is a model for T” we shall use the following: if S is a model for T and if the bounded linear operator R is similar to T, then S is a model for R.