The Structure of Inner Multipliers on Spaces with Complete Nevanlinna Pick Kernels
✍ Scribed by Devin C.V. Greene; Stefan Richter; Carl Sundberg
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 224 KB
- Volume
- 194
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
Let k be the reproducing kernel for a Hilbert space HðkÞ of analytic functions on B d , the open unit ball in C d , d51. k is called a complete NP kernel if k 0 1 and if 1 À 1=k l ðzÞ is positive definite on B d  B d . Let D be a separable Hilbert space, and consider HðkÞ D ffi Hðk; DÞ, and think of it as a space of D-valued HðkÞfunctions. A theorem of McCullough and Trent (J. Funct. Anal. 178 (2000), 226-249) partially extends the Beurling-Lax-Halmos theorem for the invariant subspaces of the Hardy space H 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 of W. Arveson (Proc. Natl Acad. Sci. USA 96 (1999), 11096-11099). # 2002 Elsevier Science (USA)