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The minimal normal extension for $$M_ z$$ on the Hardy space of a planar region. (English) Zbl 0704.47019
Let $$S=M_ z$$ be the subnormal operator defined by multiplication by the independent variable on $$H^ 2(R)$$, the space of analytic functions f on a bounded open region R in the complex plane such that $$| f|^ 2$$ has a harmonic majorant. This paper characterizes the minimal normal extension N of S. As it is known, any normal oprator is determined by a scalar-valued spectral measure and a multiplicity function. In our case the scalar-valued spectral measure for N is a harmonic measure $$\omega$$ for R. The author investigates the multiplicity function m for N. It is shown that m is bounded above by 2 $$\omega$$-a.e., and necessary and sufficient conditions are given for m to attain this upper bound on a set of positive harmonic measure. Interesting examples are discussed which indicate the relationship between N and the boundary of R.
Reviewer: M.Martin

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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