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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      .

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