Invariant subspaces for a class of complete Pick kernels

Motivated by the work of McCullough and Trent, we investigate the -invariant subspaces of the Hilbert function spaces associated to the Szego kernels on the open unit disk. In particular, we characterize those kernels for which the the -invariant subspaces are hyperinvariant, and (partially) those for which the so-called BLH subspaces are cyclic, obtaining counterexamples to two questions posed by McCullough and Trent.