Trace of Multiplication Operator Restricted to Invariant Subspaces of some Weighted Bergman Space

Let ω be a logarithmically subharmonic weight that is radial and reproducing for the origin, and L_a^2 (D,ωdA) be the weighted Bergman space. Let f be a bounded holomorphic function on the open unit disc, I be a z-invariant subspace of L_a^2 (D,ωdA), and f(M_I) denotes the restriction to I of the multiplication operator M_f. This paper investigates the trace of the self-commutator of the operator f(M_I). More precisely, we compute the trace of the commutator [f(M_I )^*,f(M_I)] and show that it equals dim⁡(I⊝zI)∫_D |f^' (z)|^2 dA(z).