The Structure of Inner Multipliers on Spaces with Complete Nevanlinna Pick Kernels

Abstract

2 ðDÞ. They show that if k is a complete NP kernel and if D is a separable Hilbert space, then for any scalar multiplier invariant subspace M of Hðk; DÞ there exists an auxiliary Hilbert space E and a multiplication operator F : Hðk; EÞ ! Hðk; DÞ such that F is a partial isometry and M ¼ FHðk; EÞ. Such multiplication operators are called inner multiplication operators and they satisfy FF * ¼ the orthogonal projection onto M. In this paper, we shall show that for many interesting complete NP kernels the analogy with the Beurling-Lax-Halmos theorem can be strengthened. We show that for almost every z 2 @B d the nontangential limit fðzÞ of the multiplier f:B d ! BðE; DÞ associated with an inner multiplication operator F is a partial isometry and that rank fðzÞ is equal to a constant almost everywhere. The result applies to certain weighted Dirichlet spaces and to the space H 2 d , which is determined by the kernel k l ðzÞ ¼ 1 1Àhz;li d ; l; z 2 B d . In particular, our result implies that the curvature invariant of Arveson (Proc. Natl. Acad. Sci. USA 96 (1999), 11,096-11,099) of a pure contractive Hilbert module of finite rank is an integer. This answers a question

Multipliers and essential norm on the Drury-Arveson space

“…Because of its connection to various topics in operator theory, e.g. the von Neumann inequality for commuting row contractions, H 2 n has been the subject of intense recent studies [1]- [7], [9], [10], [12].…”

“…Among the recent results related to multipliers, we would like to mention the following developments. Interpolation problems for multipliers and model theory related to the Drury-Arveson space have been intensely studied over the past decade or so [4,5,10,12]. Recently, Arcozzi, Rochberg and Sawyer gave a characterization of the multipliers in terms of Carleson measures for H 2 n [1].…”

Invariant subspaces for a class of complete Pick kernels

“…See [4] for details. Greene, Richter, and Sundberg [7] have shown that when Ω = B d (the unit ball of C d ) and the kernel k satisfies some mild additional assumptions, the vector-valued multiplier Φ(ζ) is a coisometry almost everywhere on the boundary of B d , strengthening the analogy with the usual Beurling theorem. In the case of the Szegő kernel, the function b is b(z, w) = zw.…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…This completes the proof. In [GRSu02], Green, Richter and Sundberg prove that for almost every ζ ∈ ∂B n the nontangential limit Θ(ζ) of the inner multiplier Θ is a partial isometry. Moreover, the rank of Θ(ζ) is equal to a constant almost everywhere.…”