Abstract
We show that there is a natural relationship between the Jensen function of certain analytic almost periodic functions h and the principal function of an associated Toeplitz operator Wh in type Π∞ von-Neumann algebra. We show that the principal function is a generalized winding number of a part of the essential spectrum of Wh with Hausdorff dimension one, and we use facts from geometric measure theory to analyze the stability of the mean motion. This is another example of our extension of the index onto a "thick" essential spectrum.
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Carey, R., Pincus, J. Mean motion, principal functions, and the zeros of dirichlet series. Integr equ oper theory 2, 484–502 (1979). https://doi.org/10.1007/BF01691074
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DOI: https://doi.org/10.1007/BF01691074