The minimal normal extension for 𝑀_{𝑧} on the Hardy space of a planar region

Abstract

Multiplication by the independent variable on H 2 ( R ) {H^2}(R) for R R a bounded open region in the complex plane C \mathbb {C} is a subnormal operator. This paper characterizes its minimal normal extension N 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 N is harmonic measure for R R , ω \omega . This paper investigates the multiplicity function m m for N N . It is shown that m m is bounded above by two ω \omega -a.e., and necessary and sufficient conditions are given for m m to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between N N and the boundary of R R .