Abstract
In a recent paper we conjectured that the principal function of a cyclic subnormal operator T is a.e. equal to the negative of a characteristic function. We showed that this was true in a variety of cases - including the general arc length Swiss Cheese.
Now we prove stronger results. The conjecture is a consequence of:
The principal function of a subnormal operator with trace class self-commutator assumes a.e. nonpositive integer values.
It is an interesting fact that this integrality is a basic geometric property of subnormal operators and is not associated with any smoothness or "thinness" of the essential spectrum of T.
This result is actually a simple corollary of a much more basic fact:
The mosaic of a subnormal operator with trace class self-commutator is projection valued a.e.
We have long known that the mosaic is a complete unitary invariant for T. Thus, this theorem establishes a map z ↦ Range B(z) which associates a subspace of Hilbert space with almost every point of the plane; and this generalized bundle completely characterizes the subnormal operator T. If T is cyclic then its mosaic B(·) is a.e. either the zero operator or a rank one projection.
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Carey, R.W., Pincus, J.D. An integrality theorem for subnormal operators. Integr equ oper theory 4, 10–44 (1981). https://doi.org/10.1007/BF01682745
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DOI: https://doi.org/10.1007/BF01682745