Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

$C^*$-Algebra $B_H(I)$ Consisting of Bessel Sequences in a Hilbert Space

中文关键词:  $C^*$-代数  Bessel列  Hilbert空间  框架  Riesz基

英文关键词:$C^*$-algebra  Bessel sequence  Hilbert space  frame  Riesz basis

中文摘要:

      设 $H$ 是 Hilbert 空间, $B_H(I)$, $B(H)$ 和 $K(H)$ 分别表示 $H$ 上的所有 Bessel 列 $\{f_i\}_{i\in I}$、有界线性算子以及紧算子之集. 利用两种等距线性同构 $\alpha_H:B_H(I)\rightarrow B(\ell^2),\beta:B_H(I)\rightarrow B(H)$ 定义了空间 $B_H(I)$ 上的两种乘法与对合运算, 使得 $B_H(I)$ 成为有单位的 $C^*$- 代数. 首先, 证明了 $C^*$-代数 $(B_H(I),\circ, \sharp)$ 与 $(B_H(I), \cdot, *)$ 是 $*$-同构的. 其次, 得到了 $H$ 上的所有框架之集 $F_H(I)$ 是有单位的乘法半群, $H$ 上的所有 Riesz 基之集 $R_H(I)$ 是自伴的乘法群, 而且, $K_H(I):=\beta^{-1}(K(H))$ 是 $B_H(I)$ 的唯一的自伴理想.

英文摘要:

      Let $H$ be a separable Hilbert space, $B_H(I)$, $B(H)$ and $K(H)$ the sets of all Bessel sequences $\{f_i\}_{i\in I}$ in $H$, bounded linear operators on $H$ and compact operators on $H$, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms $\alpha_H:B_H(I)\rightarrow B(\ell^2), \beta:B_H(I)\rightarrow B(H)$, respectively, so that $B_H(I)$ becomes a unital $C^*$-algebra under each kind of multiplication and involution. It is proved that the two $C^*$-algebras $(B_H(I), \circ, \sharp)$ and $(B_H(I), \cdot, *)$ are $*$-isomorphic. It is also proved that the set $F_H(I)$ of all frames for $H$ is a unital multiplicative semi-group and the set $R_H(I)$ of all Riesz bases for $H$ is a self-adjoint multiplicative group, as well as the set $K_H(I):=\beta^{-1}(K(H))$ is the unique proper closed self-adjoint ideal of the $C^*$-algebra $B_H(I)$.