This article, prepared for the Problem List of the Bench Math Session 2025, discusses the (anti)holomorphic rigidity phenomenon of Kobayashi and Carathéodory isometries. In particular, we provide an overview of this phenomenon, summarize several known results, and pose some open questions.
We shall discuss the characterization of cyclicity of polynomials in Dirichlet-type spaces in the Euclidean unit ball of Cn ${\mathbb{C}}^{n}$ which remains an open problem.
This note presents a problem submitted for the Bench Math Session 2025. The problem concerns the understanding of weighted Szeg & odblac; kernels in relation to auxiliary functions considered by Nehari. In particular, we pose a question about the zeros of the weighted Szeg & odblac; kernel, which lies within the scope of the classical Nehari problem studied in the literature.
We introduce the reader to partial isometries and their properties. In particular, we consider the Gau-Wang-Wu conjecture from 2016: When is the numerical range of a partial isometry circular? In order to describe a simple proof of a special case of the conjecture, we discuss basic properties of Blaschke products, compressions of the shift operator and some classical geometric methods.
In this paper, we are interested in studying surjective isometries of C*-algebras with an emphasis on their complex spectrum. We do not require the isometries to be linear nor the C*-algebras to be unital. We first characterize such isometries in terms of a Jordan & lowast;-isomorphism, a central projection, and a unitary element of the multiplier algebra, following a long line of work that began with the Banach-Stone theorem. In our main result, we then establish, for a wide class of surjective isometries, a precise connection between the complex spectrum of the isometry and the classical spectrum of the associated Jordan & lowast;-isomorphism. Finally, we turn our attention to periodic surjective isometries and provide several examples that illustrate the range of possibilities that can occur for the complex spectrum of the isometry and classical spectrum of the Jordan & lowast;-isomorphism.
