Abstract. In the category of Hilbert modules $M$ over a function algebra $A$ we introduce the notion of Nagy--Foiaş diagram which, in case it exists, connects in a special way a minimal subspectral resolution of $M$ with its corresponding minimal subspectral resolution of the adjoint module $M_{*}$ associated via the minimal spectral dilation of $M$. We show that there is a one-to-one correspondence between Nagy--Foiaş diagrams and a class of $A$-module maps. In case $A$ is the disk algebra, the Nagy--Foiaş diagram expresses the geometry of the space of the minimal unitary dilation of the contraction $T$ which generates the $A$-module structure on $M$, while the class of $A$-module maps is the class of the purely contractive analytic functions. The above correspondence is in this case the Nagy--Foiaş model based on the characteristic function.

AMS Subject Classification (1991): 47A20, 46E20

Keyword(s): Hilbert module, spectral dilation, Silov resolution, Nagy--Foiaş model

Received November 17, 1998, and in final form September 27, 1999. (Registered under 2737/2009.)