Maximal invariant subspaces in the Bergman space

Abstract

here, dS(z)=dx dy is area measure in the plane (z=x+iy). This space is usually called the Bergman space. A closed subspace 34 of A 2 is said to be invariant (or z-invariant) if z34 is contained in 34. Since the operator of multiplication by z is bounded below on A 2, z M is a closed subspace of M . We define the index of the invariant subspace 34 to be the dimension of the quotient space 3 4 / z 3 d , with values in the set {0, 1, 2 , . . . , +oc}. We will at times refer to this number as ind(34). The index of 34 can only equal 0 if 34 is the zero subspace. There are invariant subspaces of arbi t rary index [4], [6]. Let z[34] denote the operator on A2/34 induced by z, z [ M ] ( f + 3 4 ) = zf+34.