Schur class operator functions and automorphisms of Hardy algebras

By P. Muhly, B. Solel. Documenta Mathematica, 2006

52 citations recorded by the source corpus.

Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ be the associated Hardy algebra. If $\sigma$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E^{\sigma}$ and represent elements in $H^{\infty}(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^{\sigma})^{\ast}$. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of $H^{\infty}(E)$ in terms of special M\"{o}bius transformations on $\mathbb{D}(E^{\sigma})$. Particular attention is devoted to the $H^{\infty}% $-algebras that are associated to graphs.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c7ad9a10609ed995245e447c2aa128b03eb5eb9

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Citations in the corpus

Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “We mention that Muhly-Solel actually considered a much more general setting where the multiplier algebra M(kQrow ⊗ IU , kQrow ⊗ IY) is replaced by the generalized Hardy algebra H ∞ E associated with a correspondence E over a von Neumann algebra M (see also [61, 63]).”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, unknown): “11], [34], [48], [51] or [69, Theorem 2.”
Operator Theory on Noncommutative Domains (2007, unknown): “We should also remark that, in recent years, many results concerning the theory of row contractions were extended by Muhly and Solel ([54], [55], [56], [57]) to representations of tensor algebras over C∗-correspondences and Hardy algebras.”
Free holomorphic automorphisms of the unit ball of B(ℋ) n (2008, background): “We remark here that the conformal automorphisms of Bn also occur in the work of Muhly and Solel ([17]) concerning the automorphisms of Hardy algebras associated with W ∗-correspondence over von Neumann algebras ([15], [16]), and the work of Power and Solel ([33]) in a related context.”
Noncommutative reproducing kernel Hilbert spaces (2016, background): “nd the BBLS-completely positive kernels appear prominently in the characterization of the generalized Schur classes and associated generalized Nevanlinna-Pick interpolation theory of Muhly-Solel (see [34, 35, 36]). NC REPRODUCING KERNEL HILBERT SPACES 49 3.4. Smoothness properties. We nowexplore the”
Free holomorphic functions on the unit ball of B(H)n (2006, background): “Related to our work, we mention the papers [19,22,25,26,53], where several aspects of the theory of noncommutative analytic functions are considered in various settings.”
Foundations of Free Noncommutative Function Theory (2012, background): “Such classes of formal power series appear as the noncommutative generalization of the classical Schur class of contractive analytic functions on the unit disc [2, 48, 14] in the operator model theory for row contractions and more general noncommuting operator tuples [81, 82, 83, 84, 87] and in the ”