On isometric dilations of commutative systems of linear operators

Анотація

The isometric dilation of two parameter semigroup $T(n)=T^{n_1}_1 T^{n_2}_2$, where $n=(n_1,n_2)\in \mathbb{Z}^2_+$, for a commutative system $\{T_1,T_2\}$ of linear bounded operators, one of which is a contraction, $||T_1||\le 1$, is constructed. The building of the dilation is based on characteristic qualities of the commutative isometric expansion $\left\{V_s,\stackrel{+}{V_s}\right\}_{s=1}^2$, which was given in the previous work by the author [8]. The isometric dilations $U(n)$ and $\stackrel{+}{U}\!\!(n)$ of the semigroups $T(n)$ and $T^*(n)$ are shown to be unitarily linked.

Mathematical Subject Classification: 47A45.