Normal operators constructed from generalized harmonic measures on open Riemann surfaces
Hisashi Ishida · Kodai Mathematical Journal · 1993
Let R be an open Riemann surface and V be a union of a finite number of regular subregions in R with disjoint closures.We assume that R-V is connected.Denote by C ω (dV) the space of real analytic functions on dV and by H(R-V) the space of harmonic functions on R-V, A linear operatordv dv dv The notion of normal operators was introduced by L. Sario [13].He constructed two normal operators L o and L x .Here we are specially concerned with L r operator.If R is a compact bordered surface with smooth boundary, LJ is characterized by the following additional properties: i/=constant on β J9 \. *dLJ=0, where β, are the boundary components of R. For a general open Riemann surface R, LJ is defined as lim^oo L\ n f, where {R n \ is a canonical exhaustion of R and L?» is the L r oρerator from C ω (dV) to H(R n - V).Let Γ h (R) be the Hubert space of real square integrable harmonic differentials on R and Γ h 8e(R) be the space of semiexact differentials in Γ h (R).Let us denote by Γ hm (R) the orthogonal complement of *Γ hse (R) in L h {R).Then LJ