Complete Nevanlinna-Pick kernels

By Jim Agler, John E. McCarthy. 2000

182 citations recorded by the source corpus.

We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/443c2e922bf9e9b9123beb98fa643a9c74ae9145

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Citations in the corpus

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, background): “ property, i.e., the interpolation question for matrix valued functions has a satisfactory answer (see [11,65] and [2, Section 8.9] and the references therein). Moreover, Agler and McCarthy proved in [1] that this space is universal among the spaces with the complete Pick property. 2 Let d2N [f1g, a”
Hilbert spaces of Dirichlet series and their multipliers (2003, methodology): “As the reciprocal of k has only one positive square, it follows from the McCulloughQuiggin theorem (see [11], [14], [1]), that k is a complete Pick kernel.”
The Dirichlet space: a survey (2010, background): “We redirect the interested reader to the book [2], to the article [3] and to the important manuscript [37] itself for far reaching developments of this line of reasoning.”
Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis (2013, methodology): “ems (recall that a linear system can be seen as a holomorphic or a rational function dened in the unit disk and whose values are matrices). Analogues of Schur functions were introduced by Agler, see [1, 2]. Besides the Schur-Agler classes, we mention that counterparts of Schur functions have been st”
Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls (2007, background): “12 of [2] shows that the reproducing kernel k (z, w) = ( 1 1−w·z )2σ has the complete Nevanlinna-Pick property.”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “(2) in Theorem 5.5 to the case E = Y, a(Z) = IYn for Z∈ Ωn yields the following corollary. Corollary 5.6. The kernel kQ0 given by (5.3) is a complete Pick nc kernel. Remark 5.7. It is well known (see [72, 4] as well as [5, Theorem 7.6]) that in the classical case the Szego˝ kernel has a certain univ”
Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in Cn (2004, methodology): “The proof of ð2Þ ) ð5Þ is an adaptation of the ‘‘lurking isometry’’ method which has been used recently as a method of proof in a number of closely related realization and interpolation problems for multivariable functions (see [4,8,11–13,15,19] and also [33–35] for an abstract formalism in the sing”
On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces (2015, methodology): “ so if V = {zn : n∈ N}, then Mult(H2 V ) and Mult(D V ) are equal as algebras, since they are both equal to ℓ∞. In fact, the normalized kernels in D V and H2 V form a Riesz system (see Section 9.3 in [1]), so there is a bounded invertible map A: H2 V → D V such that A K H2(·,w) ||KH2(·,w)|| = KD(·”
Operator algebras for analytic varieties (2012, background): “2 in [1], every irreducible complete Nevanlinna-Pick kernel is equivalent to the restriction of the kernel of Drury-Arveson space to a subset of the ball.”
Complete Nevanlinna-Pick property of Dirichlet-type spaces (2002, background): “Moreover, as shown in [1], any reproducing kernel in the form”
A Constrained Nevanlinna-Pick Interpolation Problem (2007, methodology): “he positivity of the matrix (A2Ik − WiW∗ j )K(zi,zj) is equivalent to interpolation, the kernel is called a complete Nevanlinna–Pick kernel. Such kernels have been characterized by Agler and McCarthy [2]. However, recognition of such kernels is generally not straightforward. In our context of H∞ 1 ,”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “ion. Theorem 7.2.4. The Drury-Arveson space H2 d has the complete Pick property. Proof. This theorem has several proofs. A Hilbert function space theoretic proof was given by J. Agler and J. McCarthy [2] (following works of McCullough [90] and Quiggin [104]). In fact [2] characterizes all complete P”
Multiplier algebras of complete Nevanlinna–Pick spaces: Dilations, boundary representations and hyperrigidity (2016, background): “llowing precise sense. The C∗-envelope of A is a C∗- algebra C∗ e (A) together with a unital completely isometric linear map ι: A → C∗ e(A) with the following two properties: (1) C∗ e (A) = C∗(ι(A)), (2) whenever j: A → B is another unital completely isometric linear map and B = C∗(j(A)), then there”