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https://openscholarship.wustl.edu/iwota2016/special/Functionspaces
Recent Events in Function Spacesen-usTue, 11 Apr 2023 16:58:24 PDT3600Nevanlinna-Pick interpolation problem in the ball
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/6
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/6Thu, 21 Jul 2016 17:30:00 PDT
We solve a three point Nevanlinna-Pick problem in the Euclidean ball. In particular, we determine a class of rational functions that interpolate this problem. This is joint work with W. Zwonek.
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Lukasz KosiĆskiTests for complete $K$-spectral sets
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/5
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/5Thu, 21 Jul 2016 17:00:00 PDT

We say that a domain $D$ in the complex plane is a spectral set for $T$, a bounded operator on a Hilbert space $H$, if a form of von Neumann's inequality holds; that is, if for any rational function $f$ with poles off of $D$, $\|f(T)\| \leq \|f\|$, where the left hand norm is the operator norm of $f(T)$ and the right hand norm is the supremum norm of $f$ over $D$. It is called a complete spectral set if the inequality also holds for $f$ matrix valued for any size square matrix. The classical von Neumann inequality says that the unit disk is a (complete) spectral set for any operator with its spectrum in this set. If one tries to extend this result to more complicated domains, the attempt often fails. One potential way around this is to relax the inequality by requiring instead that $\|f(T)\| \leq K \|f\|$ for a constant $K$. Following in the footsteps of Mascioni, Stessin, Stampfli, Badea-Beckerman-Crouzeix and others, we consider planar domains that are comprised of the intersection of level sets of moduli of complex functions under certain regularity conditions. We also consider applications to a variation of the rational dilation problem. The key tool is our work on the relationship between the generation of uniform algebras of analytic functions in planar domains and the extension to the polydisk of a bounded analytic function defined on an analytic variety inside the polydisk. This is joint work with Dmitry Yakubovich and Daniel Est\'evez.

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Michael DritschelHorn inequalities for singular values of products of operators
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/4
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/4Thu, 21 Jul 2016 16:00:00 PDT
Consider Hermitian matices $A, B, C$ such that $A + B = C$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This conjecture was proved true by Klyachko and Knutson-Tao in the late 1990s. A related question, the multiplicative Horn problem, asks to describe the possible singular values of $AB$ when the singular values of $A$ and $B$ are given. This problem is fully solved in the case where $A$ and $B$ are invertible matrices as Klyachko showed that it is equivalent to the additive problem after taking logarithms. In this talk we will discuss the case when $A$ and $B$ are not necessarily invertible and its generalization to the von Neumann algebra setting. This is joint work with H. Bercovici, B. Collins, and K. Dykema.
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Wing Suet LiWeighted composition operators from Banach spaces of analytic functions into Bloch-type spaces
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/3
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/3Thu, 21 Jul 2016 15:30:00 PDT
Let $X$ be a Banach space of analytic functions on the unit disk $\mathbb D$ whose point evaluation functionals are continuous. We study weighted composition operators from $X$ into Bloch type spaces. Imposing certain natural conditions on $X$ we are able to characterize all at once the bounded and the compact operators as well as in many cases give estimates or precise formulas for the essential norm. One condition used is: \medskip \noindent (VI) \ There exists $C>0$ such that $\|Sf\|\le C\|f\|$, for all $f$ in $X$ and for all disk automorphisms $S$. \smallskip \noindent When $X$ is either the Bloch space or the space of analytic functions, $S^p$, whose derivatives are in the Hardy space $H^p$ though, (VI) fails. So when $X$ is continuously contained in the Bloch space, we impose two other conditions on the norm of the point evaluation functionals. In the end our results apply to known spaces that include the Hardy spaces, the weighted Bergman spaces, $BMOA$, the Besov spaces and all spaces $S^p$. This is joint work with Flavia Colonna.
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Maria TjaniThe Dirichlet Space on the bi-disc
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/2
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/2Thu, 21 Jul 2016 15:00:00 PDT
We present some preliminary findings on properties of the Dirichlet space on the bi-disc, which is defined as the tensor product of two copies of the classical Dirichlet space on the unit disc. Work in collaboration with Pavel Mozolyako, Karl-Mikael Perfeckt, Giulia Sarfatti.
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Nicola ArcozziAlgebraic and geometric aspects of rational $\Gamma$-inner functions
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/1
https://openscholarship.wustl.edu/iwota2016/special/Functionspaces/1Thu, 21 Jul 2016 14:30:00 PDT
The set \[ \Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\"obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$. The talk is based on joint work with Jim Agler and Nicholas Young. \begin{itemize} \item[{[1]}] Jim Agler, Zinaida A. Lykova and N. J. Young: Algebraic and geometric aspects of rational $\Gamma$-inner functions, (arXiv: 1502.04216 [math.CV] 17 Febr. 2015) 22 pp. \end{itemize}
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Zinaida Lykova