Abstract. We prove a structure theorem for a finite set G of partial isometries in a fixed countably infinite dimensional complex Hilbert space H. Our result is stated in terms of the C * -algebra generated by G. The result is new even in the case of a single partial isometry which is not an isometry or a co-isometry; and in this case, it extends the Wold decomposition for isometries. We give applications to groupoid C * -algebras generated by graph groupoids, and to partial isometries which have finite defect indices and which parametrize the extensions of a fixed Hermitian symetric operator with dense domain on the Hilbert space H. Our classification parameters for the "Wold decomposed" set G W of our finite set G of partial isometries involve infinite and explicit Cartesian product sets, and they are computationally attractive. Moreover, our classification labels generalize the notion of defect indices in the special case of the family G W of partial isometries from the Cayley transform theory and Hermitian extensions of unbounded Hermitian operators with dense domain.

In this paper, we use explicit formulas for the moments of a self-adjoint operator (radial operator), induced by a certain discrete structure, in Hilbert space. Our main theorem shows that the structures are classified by the moment computations producing an equivalence relation. Our motivation in turn derives from a groupoidtheoretic approach to spectral problems as they arise in quantum mechanics. While it is typically difficult to obtain explicit formulas for spectra, we demonstrate that our moment formulas serve as a substitute. The discrete structures G we study have a built in fractal feature: any portion of G is similar to the whole. And this fact (like in renormalization groups) serves to facilitate computations.

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