On Toeplitz-Invariant Subspaces of the Bergman Space

A subspace M ⊂ L2a(Δ) = A2, is called an e-subspace if (i) dim M < ∞; (ii) 1 ∈ M; (iii) M ⊂ H∞; (iv) for every ƒ ∈ A2, such that (|ƒ|2 − 1) is orthogonal to M, and every g ∈ M, ||fg|| ≥ ||g||. Define the operator T by (Tƒ)(z) = ∫Δ |ƒ(w)|2K(z, w)dA(w), where K(z, w) = (1/π)(l − zw)−2 is the Bergman kernel in Δ. A subspace M ⊂ A2 satisfying (i), (ii), (iii) is called a T-subspace if TM ⊂ M. It is proved that M is an e-subspace if and only if M is a T-subspace. In particular, a finite dimensional linear space M of polynomials is an e-subspace if and only if M = span{zkj}Nj = 0 where k > 0 and N ≥ 0 are integers. For k = 1 this implies a sharper form of a theorem of H. Hedenmalm.