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

By Devin C. V. Greene, Stefan Richter, C. Sundberg. 2001

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Let k be the reproducing kernel for a Hilbert space H(k) of analytic functions on Bd, the open unit ball in Cd, d⩾1. k is called a complete NP kernel if k0≡1 and if 1−1/kλ(z) is positive definite on Bd×Bd. Let D be a separable Hilbert space, and consider H(k)⊗D≅H(k,D), and think of it as a space of D-valued H(k)-functions. 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 H2(D). 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 H(k,D) there exists an auxiliary Hilbert space E and a multiplication operator φ: H(k,E)→H(k,D) such that φ is a partial isometry and M=φH(k,E). Such multiplication operators are called inner multiplication operators and they satisfy φφ;*=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∈∂Bd the nontangential limit φ(z) of the multiplier φ:Bd→B(E,D) associated with an inner multiplication operator φ is a partial isometry and that rank φ(z) is equal to a constant almost everywhere. The result applies to certain weighted Dirichlet spaces and to the space H2d, which is determined by the kernel kλ(z)=11−〈z,λ〉d ,λ,z∈Bd. In particular, our result implies that the curvature invariant of Arveson (Proc. Natl. Acad. Sci. USA96 (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. USA96 (1999), 11096–11099).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/71bbf88fb5feb095a3a611ca3b7b633599b08aa6

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Citations in the corpus

On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces (2015, background): “y observing that the kernel of a Nevanlinna-Pick space can be recovered from the isometric structure of its multiplier algebra. Results similar to the next proposition are well known, see for example [22] and [3], especially Exercise 8.35. Since we do not have a reference for the exact statement, a ”
Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem (2002, background): “One knows (see [ 22 ], [25]) that subspaces of H(k) invariant under multiplication by the independent variables are of the form BH(k) TM, where n can be co, and B is an inner function in some sense.”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “arger regions of convergence, see [107, Theorem 5.6.4]). Arveson raised the problem of whether or not P n|φ˜n(w)|2 = 1 for a.e. w∈∂Bd. This problem was solved by D. Greene, S. Richter and C. Sundberg [73]. For every λ∈Bd, let Eλ: H2 d⊗K→Kdenote the point evaluation functional Eλf⊗k= f(λ)k. Theorem 8”
p-Summable Commutators in Dimension d (2003, result): “ils to hold in general. Still, that formula provided enough evidence to lead us to conjecture that K(T¯) is an integer in general, and that has now been established by Greene, Richter and Sundberg in [GRS02]. Unfortunately, the integer arising in the proof of the latter result - namely the rank of a”
Quotients of standard Hilbert modules (2005, background): “ture on various aspects of the theory commuting d-tuples of operators on Hilbert spaces – i.e., Hilbert modules over C[z1,...,zd]. A sample follows: [CD78], [Dou05], [DMV00], [DMV01], [DM03], [DP89], [GRS02], [GS05], [MV93], [Tay70a], [Tay70b], [Vas92]. Finally, it seems appropriate to point out tha”
The Dirac Operator of a Commuting d-Tuple (2000, result): “By the results of [ 9 ], the codimension of M(z) ⊆ Cr+1 is equal to K(¯”
Conservative Structured Noncommutative Multidimensional Linear Systems (2005, result): “We mention [1, 2, 17, 15, 16] for extensions to the polydisk D n ⊂ C n setting, [33, 52, 10, 30, 3, 8, 18, 45, 37 , 4 ] for extensions to the unit ball B n ⊂ C n (where additional refinements concerning Nevanlinna-Pick-type interpolation and lifting theorems are also explored) ,a nd the recent work”