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Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilationto a Hardy shift and thus established an elegant analytic functional model forcontractions of class $C_{\cdot0}$. This has motivated lots of further works onmodel theory and generalizations to commuting tuples of$C_{\cdot0}$-contractions. In this paper, we focus on doubly commutingsequences of $C_{\cdot0}$-contractions, and establish the dilation theory andthe analytic model theory for these sequences of operators. These results areapplied to generalize the Beurling-Lax theorem and Jordan blocks in themultivariable operator theory to the operator theory in countably infinitelymany variables.

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions