A Class of Operators on Lh 2

Let D be the open unit disc in ? 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,x?Dφ(x) for all b ? ?D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ? 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.