Inner sequence based invariant subspaces in 𝐻²(𝐷²)

Abstract

A closed subspace H 2 ( D 2 ) H^{2}(D^2) is said to be invariant if it is invariant under the Toeplitz operators T z T_z and T w T_w . Invariant subspaces of H 2 ( D 2 ) H^{2}(D^2) are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for ∑ j 1 − | λ j | \sum _j 1-|\lambda _j| , where { λ j } \{\lambda _j\} are zeros of a Blaschke product.