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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The minimal normal extension for $ M\sb z$ on the Hardy space of a planar region


Author: John Spraker
Journal: Trans. Amer. Math. Soc. 318 (1990), 57-67
MSC: Primary 47B20; Secondary 47B15, 47B38
DOI: https://doi.org/10.1090/S0002-9947-1990-1008703-3
MathSciNet review: 1008703
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Abstract: Multiplication by the independent variable on $ {H^2}(R)$ for $ R$ a bounded open region in the complex plane $ \mathbb{C}$ is a subnormal operator. This paper characterizes its minimal normal extension $ N$. Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for $ N$ is harmonic measure for $ R$, $ \omega $. This paper investigates the multiplicity function $ m$ for $ N$. It is shown that $ m$ is bounded above by two $ \omega $-a.e., and necessary and sufficient conditions are given for $ m$ to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between $ N$ and the boundary of $ R$.


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DOI: https://doi.org/10.1090/S0002-9947-1990-1008703-3
Keywords: Harmonic measure, multiplication operator, minimal normal extension
Article copyright: © Copyright 1990 American Mathematical Society