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On quasi-free Hilbert modules. (English) Zbl 1098.46019
Hilbert modules are Hilbert spaces that are acted upon by a natural algebra of functions holomorphic on some bounded domain in complex \(n\)-space \(\mathbb C^n\). They are useful in the study of multivariate operator theory.
In the paper under review the authors investigate quasi-free Hilbert modules which are the Hilbert space completion of a space of vector-valued holomorphic functions that possesses a kernel function; cf. R. G. Douglas and G. Misra [Integral Equations Oper. Theory 47, No. 4, 435–456 (2003; Zbl 1073.47502)]. They examine the bundle associated with a quasi-free module and introduce a nonnegative matrix-valued modulus function for any pair of finite rank quasi-free Hilbert modules and give a necessary condition for two such modules to be unitarily equivalent.

MSC:
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
32B99 Local analytic geometry
32L05 Holomorphic bundles and generalizations
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