Invariant subspaces in Banach spaces of analytic functions

Abstract:

We study the invariant subspace structure of the operator of multiplication by $z$, ${M_z}$, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class ${\mathcal {B}_1}(\overline \Omega )$. We say that an invariant subspace $\mathcal {M}$ satisfies $\operatorname {cod} \mathcal {M} = 1$ if $z\mathcal {M}$ has codimension one in $\mathcal {M}$. We give various conditions on invariant subspaces which imply that $\operatorname {cod} \mathcal {M} = 1$. In particular, we give a necessary and sufficient condition on two invariant subspaces $\mathcal {M}$, $\mathcal {N}$ with $\operatorname {cod} \mathcal {M} = \operatorname {cod} \mathcal {N} = 1$ so that their span again satisfies $\operatorname {cod} (\mathcal {M} \vee \mathcal {N}) = 1$. This result will be used to show that any invariant subspace of the Bergman space $L_a^p, p \geqslant 1$, which is generated by functions in $L_a^{2p}$, must satisfy $\operatorname {cod} \mathcal {M} = 1$. For an invariant subspace $\mathcal {M}$ we then consider the operator $S = M_z^{\ast }|{\mathcal {M}^ \bot }$. Under some extra assumption on the domain of holomorphy we show that the spectrum of $S$ coincides with the approximate point spectrum iff $\operatorname {cod} \mathcal {M} = 1$. Finally, in the last section we obtain a structure theorem for invariant subspaces with $\operatorname {cod} \mathcal {M} = 1$. This theorem applies to Dirichlet-type spaces.