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.

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In this paper we propose a different (and equivalent) norm on S 2 (D) which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of S 2 (D) in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of C ϕ for a class of composition operators. We completely characterize multiplication operators which are m-isometries. As an application of the 3-isometry, we describe the reducing subspaces of M ϕ on S 2 (D) when ϕ is a finite Blaschke product of order 2.

We investigate expansive Hilbert space operators T that are finite rank perturbations of isometric operators. If the spectrum of T is contained in the closed unit disc D, then such operators are of the form T = U ⊕ R, where U is isometric and R is unitarily equivalent to the operator of multiplication by the variable z on a de Branges-Rovnyak space H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim ker T * = 1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B| 2 + |a| 2 = 1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z) = a(0) p(z) q(z) , where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w ∈ ∂D.