Hankel operators and invariant subspaces of the Dirichlet space

Abstract

The Dirichlet space D is the space of all analytic functions f on the open unit disc D such that f is square integrable with respect to two-dimensional Lebesgue measure. In this paper, we prove that the invariant subspaces of the Dirichlet shift are in one-to-one correspondence with the kernels of the Dirichlet-Hankel operators. We then apply this result to obtain information about the invariant subspace lattice of the weak product D D and to some questions about approximation of invariant subspaces of D.Our main results hold in the context of superharmonically weighted Dirichlet spaces.

Hankel operators, invariant subspaces, and cyclic vectors in the Drury-Arveson space

“…We note that this is analogous to Theorem 1.2 of [10], where the corollary is proved for the Dirichlet space.…”

“…We mentioned before that it follows from Beurling's theorem that every invariant subspace of the unilateral shift (M z , H 2 ) equals the kernel of a bounded Hankel operator. Similarly, it was shown in [10] that every invariant subspace M of the Dirichlet shift (M z , D) satisfies M = ker H b for some b ∈ X (D). For the Bergman space no direct analog of such a theorem can hold (see [18]).…”

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

“…However, for many interesting expansive operators T the defect operator ∆ = T * T −I will be compact. That is true for example for the Dirichlet shift, and more generally, if T = M z on a superharmonically weighted Dirichlet space (see [29], Theorem 5.1). If this is the case, then ∆ = n≥1 t n f n ⊗ f n for some 0 < t n → 0 and an orthonormal basis {f n } of ran ∆, and one can show that for the B one can take b n /z = tn 1+tn f n .…”

Multipliers and operator space structure of weak product spaces

“…the introduction of [4]. Weak product spaces have been concretely studied for instance for the classical Dirichlet space [4,12,16], the Drury-Arveson space [17] and more generally for complete Nevanlinna-Pick spaces [3,10].…”