A Class of Operators on L_h ²

abstract

Let D be the open unit disc in ℂ and let L_h ² be the space of quadratic integrable harmonic functions defined on D. Let $\varphi: {\bar D}\rightarrow {\rm C}$ be a function in L^∞(D) with the property that φ(b) = lim_x→b,xϵDφ(x) for all b ϵ ∂D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ϵ L_h ², where Q is the orthogonal projection from L² (D) on L_h ². The following results are proved. If φ¦_∂D ≡ 0, then C_φ is a compact linear operator and if φ¦_∂D vanishes nowhere, then C_φ is a Fredholm operator.