Isometric expansions of commutative systems of linear operators

V. A. Zolotarev

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 3, pp. 282-301

Résumé

The commutative isometric expansion $\bigl\{V_s,\stackrel{+}{V_s}\bigr\}_{s=1}^2$ for a commutative system $\left\{T_1,T_2\right\}$ of linear bounded operators in Hilbert space $H$ is constructed. Building of the isometric dilation for two parameter semigroup $T(n)=T_1^{n_1}T_2^{n_2}$, where $n=(n_1;n_2)$, is based on characteristic qualities of given commutative isometric expansion. Main properties of a characteristic function $S(z)$, corresponding to the commutative isometric expansion $\bigl\{V_s,\stackrel{+}{V_s}\bigr\}_{s=1}^2$ are described. An analogue of Hamilton–Cayley theorem is proved. It is shown that there exists polynomial $\mathbb{P}(z_1,z_2)$ such as $\mathbb{P}(T_1,T_2)=0$ when the defect subspaces of system $\{T_1,T_2\}$ are of finite dimension.