The Brown analysis seminar typically meets on Mondays at 3pm in Kassar 105.
Analysis Seminar
Current schedule of talks, Spring 2025
January 27: Sergei Treil, Brown University
Title: The inverse spectral problem for positive Hankel operators
Abstract: Hankel operators are bounded operators in \ell^2 =\ell^2(\mathbb Z_+) with matrix of form \{\gamma_{j+k}\}_{j,k\ge0} . They can be realized as the integral operators in L^2(\mathbb R_+),
\displaystyle \Gamma f(s) =\int_0^\infty h(s+t) f(t) dt.
We consider positive semidefinite Hankel operators with simple spectrum and solve the inverse spectral problemfor such operators, i.e. reconstruct coefficients \gamma_k or the kernel h from the appropriately chosen spectral measure (and some other spectral invariants). It turns out that solutions of the spectral problem are drastically different for discrete and continuous realizations: to reconstruct the kernel h one only needs the natural (for the problem) spectral measure, for the coefficients \gamma_k one needs extra spectral invariant.
The talk is based on a joint work with A. Pusnitski.
February 3: Constanze Liaw, University of Delaware
Title: An Overview of Aleksandrov-Clark Theory and some Generalizations
Abstract: We will begin by recalling the origination of Aleksandrov-Clark Theory: First note that Beurling’s Theorem says that any non-trivial shift-invariant subspace of the Hardy space $H^2(\mathbb{D})$ is of the form $\theta H^2(\mathbb{D})$ for an inner function $\theta$. Now, for a fixed inner $\theta$, we form the model space, that is, the orthogonal complement of the corresponding shift-invariant subspace in the Hardy space. Consider the compressed shift, which is the application of the shift to functions from the model space followed by the projection to the model space. Clark observed that all rank-one perturbations of the compressed shift that are also unitary have a particular, simple form. Following this discovery, a rich theory was developed connecting the spectral properties of those unitary rank one perturbations with properties of functions from the model space, more precisely, with their non-tangential boundary values. Some intriguing perturbation results were obtained via complex function theory.
Throughout we will allude to some generalizations such as non-inner $\theta$, finite rank perturbations, functions of several variables, etc., and we will also focus on some ongoing work on the bidisk.
March 17: Nathan Wagner, Brown University
Title: TBA
Abstract: TBA
April 14: Lukas Bundrock, University of Alabama
Title: TBA
Abstract: TBA
April 21: Jose Madrid Padilla, Virginia Tech
Title: TBA
Abstract: TBA
April 28: Brandon Sweeting, Washington University in St. Louis
Title: TBA
Abstract: TBA
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If you have questions or comment, please contact the organizers: Benoit Pausader, Jill Pipher and Nathan Wagner.