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Creators/Authors contains: "Augat, Meric"

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  1. ; ; ; (, Proceedings of the American Mathematical Society)
    Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial f f in d d freely noncommuting arguments, find a free polynomial p n p_n , of degree at most n n , to minimize c n ≔ ‖<#comment/> p n f −<#comment/> 1 ‖<#comment/> 2 c_n ≔\|p_nf-1\|^2 . (Here the norm is the ℓ<#comment/> 2 \ell ^2 norm on coefficients.) We show that c n →<#comment/> 0 c_n\to 0 if and only if f f is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the d d -shift. 
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  2. ; (, Canadian Mathematical Bulletin)
    Abstract We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g.,$$B({\mathcal H})$$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the$$k{\mathrm {th}}$$directional derivative of any NC function at a scalar point is ak-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball. 
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  3. ; ; (, Journal of Mathematical Analysis and Applications)
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