## The minimal normal extension for $M_ z$ on the Hardy space of a planar region

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- by John Spraker PDF
- Trans. Amer. Math. Soc.
**318**(1990), 57-67 Request permission

## 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 $R$.## References

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## Additional Information

- © Copyright 1990 American Mathematical Society
- 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