The minimal normal extension for \(M_ z\) on the Hardy space of a planar region.

*(English)*Zbl 0704.47019Let \(S=M_ z\) 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 |