Pick Interpolation and Hilbert Function Spaces

By Jim Agler, John E. McCarthy. 2002

700 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/cc20c86bb3f69a3534887a4572219f0c852b201e

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, background): “These concepts were later given a fresh approach from the operator theoretic perspective by Sarason in [75] and by others (see [2]).”
Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls (2007, methodology): “It was pointed out to us by John McCarthy that by using Corollary 13 together with techniques from Chapter 14 of [3] it is possible to prove dilation and extension theorems for operators T which have spectrum in R̄ and which satisfy the operator inequality”
Composition operators on weighted Bergman spaces of a half-plane (2009, background): “By the assumption that K2 n is positive, and since adding a positive constant to a kernel will certainly keep it positive, this is the product of two positive kernels and K2 n+1 is therefore positive by the Schur Product Theorem [1].”
Distinguished varieties (2005, background): “It is known [1, 4] that studying all the admissible kernels on a set is essential to understanding the Pick problem.”
Norm preserving extensions of holomorphic functions from subvarieties of the bidisk (2004, unknown): “If λ1, . . . , λn are n distinct points in D 2 and z1, . . . , zn ∈ C then there exists ϕ ∈ H∞(D2) with ‖ϕ‖D2 ≤ 1 and ϕ(λi) = zi for each i if and only if (2.3) [ (1− zizj) ki(j) ] ≥ 0, for all k ∈ Kλ.”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “This shows that point evaluation is a bounded functional on H 2 d , so H 2 d is a Hilbert function space [3], and it has the kernel (3).”
On the index of invariant subspaces in spaces of analytic functions of several complex variables (2005, background): “In [8] it is shown that every pure d-contraction is the compression of Mz;H 2 d ðDÞ to the orthocomplement of some scalar multiplier invariant subspace of H 2 d ðDÞ (see also [1] and [2], Section 14.”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “[3] for a treatment in the case of the unit disk.”
Noncommutative reproducing kernel Hilbert spaces (2016, methodology): “r to develop further the work of Agler-McCarthy [2, 3] on noncommutative Pick interpolation and transfer-function realization and introduce a free noncommutative version of “complete Pick kernel”(see [1] for the classical case), as well as develop the interpolation and transfer-function realization ”
Recent Progress on Truncated Toeplitz Operators (2011, background): “n Truncated Toeplitz Operators 3 on Kzn) and the backward shift on the span of a ﬁnite collection of Cauchy kernels (i.e., Au z on a ﬁnite dimensional model space Ku). We refer the reader to the text [1] which treats these problems in greater detail and generality. Toeplitz matrices, which can be vi”
Operator algebras for analytic varieties (2012, methodology): “ows us to write 1 1− hh(z),h(w)i = X∞ n=0 X∞ i=1 |bi|2ziwi) n = X∞ n=0 an(zw)n. By a basic result in RKHSs, ˜k(z,w) = P en(z)en(w) where {en} is an orthonormal basis for k˜ (see Proposition 2.18 of [2]). Hence H is the space of holomorphic functions on D with orthonormal basis { √ anzn}∞ n=0. The ma”