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 Lₐ² (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 Lₐ² (D, ωdA), and f (MI) denotes the restriction to I of the multiplication operator Mf. This paper investigates the trace of the self-commutator of the operator f (MI). More precisely, we compute the trace of the commutator f (MI) ^*, f (MI) and show that it equals dim⁡ (I⊝zI) ∫D |f^' (z) |² dA (z).