Dilations, wandering subspaces, and inner functions

Abstract

Abstract. The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces H K on the unit ball in C n , wandering subspaces for restrictions of the multiplication tuple M z = (M z1 , . . . , M zn ) can be described in terms of suitable H K -inner functions. We prove that H K -inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way we prove a refinement of a result of Arveson on the uniqueness of minimal dilations of pure row contractions.

Factorizations induced by complete Nevanlinna–Pick factors

“…In Section 4 we show that multiplier-invariant subspaces in H k are generated by functions in Mult(H s , H k ), that extremal functions in H k (E) belong to Mult(H s , H k (E)), and derive a pointwise estimate for these functions. These recover some results in [19] and in the recent paper [8], but apply to other situations as well. The Sarason function of an extremal element is constant equal to 1 in Ω, and motivated by this observation we continue the investigation of Mult(H s , H k (E)) in terms of this object.…”

A Beurling–Lax–Halmos theorem for spaces with a complete Nevanlinna–Pick factor

“…The paper [20] further extends this result to the unit ball in C d , where the role of H 2 is played by the Drury-Arveson space. This last generalization, along with a uniqueness statement, is also obtained in [10] as part of a wider investigation of dilations and wandering subspaces.…”

Wandering subspace property for homogeneous invariant subspaces

“…where we regard E 0 ⊂ H 2 n (E 0 ) as the closed subspace consisting of the constant functions. By Theorem 5.2 in [5] the map Π 0 induces a unitary operator…”

“…By a well known invariant subspace result (see e.g. Theorem 4.1 in [5]) there are a Hilbert space E 0 and a surjective partial isometry Π 0 ∈ B(H 2 n (E 0 ), H) with Π 0 M z i = T i Π 0 for i = 1, . .…”

“…If the unit disc is replaced by the unit ball in B ⊂ C n , then the situation becomes more involved. It is known that in the standard analytic functional Hilbert spaces on the unit ball such as the Drury-Arveson space, the Hardy or Bergman space, there exist invariant subspaces of infinite index and that each of these spaces contains invariant subspaces that are not generated by their wandering subspace [5].…”