Comparing Two Generalized Noncommutative Nevanlinna–Pick Theorems

Norton, Rachael M.

doi:10.1007/s11785-016-0540-9

Cited by 2 publications

(4 citation statements)

References 11 publications

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“…We mention that Muhly-Solel actually considered a much more general setting where the multiplier algebra M(k Qrow ⊗ I U , k Qrow ⊗ I Y ) is replaced by the generalized Hardy algebra H ∞ E associated with a correspondence E over a von Neumann algebra M (see also [61,63]). An interesting topic for future research is to get a better understanding about how this general von Neumann-algebra correspondence setting fits with the free nc-function setting used here; preliminary steps in such a program have already been made in [16,65,67]. 6.3.…”

Section: 2mentioning

confidence: 99%

“…For the case of commuting variables, nc-function point-evaluation (or Riesz-Dunford) interpolation conditions can be reduced to the older theory of LTOA interpolation conditions and one can recover the solution criterion of one from the solution criterion for the other; this point is explored in more detail in [24]. The recent paper of Norton [67] explores similar connections between the interpolation theory of Constantinescu-Johnson [32] and that of Muhly-Solel [62].…”

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confidence: 99%

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Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Ball¹,

Marx²,

Vinnikov

2018

Operator Theory in Different Settings and Related Applications

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The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur-Agler-class theory to the free noncommutative function setting. This includes the positive-kernel-decomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur-Agler class coincides with the contractive-multiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.1991 Mathematics Subject Classification. 47B32; 47A60.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Ball¹,

Marx²,

Vinnikov

2018

Operator Theory in Different Settings and Related Applications

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“…Consider The following simple example, brought to our attention by the referee of [Nor17], illustrates a case in which the converse fails. Now consider two problems:…”

Section: Chapter 4 Comparison With Muhly-solel's Generalizedmentioning

confidence: 99%

“…A close study of the Pick matrices shows that interpolation in the sense of Muhly-Solel's theorem implies interpolation in the sense of Theorem 3.0.1. However, a simple example, brought to our attention by the referee of [Nor17], illustrates that the converse is not true in general. Nevertheless, in Theorem 4.1.6 we identify commutativity assumptions under which the two theorems yield the same result.…”

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confidence: 95%

Pick interpolation, displacement equations, and W*-correspondences

Norton¹

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There are many people, including officemates, classmates and their spouses, roommates, neighbors, and the mathletes, who have made Iowa City feel like home for the past six years. In particular, I would like to thank Christine, Julia, Catie, Mario, Nathaniel, Colin, Kevin, and Ze for their friendship. I feel lucky to have been surrounded by such wonderful people. Finally, I am grateful to my mom, Nancy, my dad, Mike, my sister, Emily, and my partner, Nate, for their unwavering support and an earnest interest in my work. Without your encouragement, I might have given up a long time ago.

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Comparing Two Generalized Noncommutative Nevanlinna–Pick Theorems

Abstract: We explore the relationship between two noncommutative generalizations of the classical Nevanlinna-Pick theorem: one proved by Constantinescu and Johnson in 2003 and the other proved by Muhly and Solel in 2004.

Cited by 2 publications

References 11 publications

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Pick interpolation, displacement equations, and W*-correspondences

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