# Past Functional Analysis Seminar

We consider a vector-valued function $f: \mathbb{R}_+ \to X$ which is locally of bounded variation and give a decay rate for $|A(t)|$ for increasing $t$ under certain conditions on the Laplace-Stieltjes transform $\widehat{dA}$ of $A$. For this, we use a Tauberian condition inspired by the work of Ingham and Karamata and a contour integration method invented by Newman. Our result is a generalisation of already known Tauberian theorems for bounded functions and is applicable to Dirichlet series. We will say something about the connection between the obtained decay rates and number theory.

Let $\mathrm{BMOA}_{\mathcal{NP}}$ denote the space of operator-valued analytic functions $\phi$ for which the Hankel operator $\Gamma_\phi$ is $H^2(\mathcal{H})$-bounded. Obtaining concrete characterizations of $\mathrm{BMOA}_{\mathcal{NP}}$ has proven to be notoriously hard. Let $D^\alpha$ denote differentiation of fractional order $\alpha$. Motivated originally by control theory, we characterize $H^2(\mathcal{H})$-boundedness of $D^\alpha\Gamma_\phi$, where $\alpha>0$, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that $\mathrm{BMOA}_{\mathcal{NP}}$ is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of $\Gamma_\phi$ . The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.

Given a locally compact group G, the group C*-algebra is defined by taking the completion of $L^1(G)$ with respect to the C*-norm given by the irreducible unitary representations of G. However, if the group is not abelian, there is no known concrete description of its group C*-algebra. In my talk, I will briefly introduce the group C*-algebras and then give some examples arisen from solvable Lie groups

Our long term plan is to develop a unified approach to prove decomposition theorems in different structures. In our anti-dual pair setting, it would be useful to have a tool which is analogous to the so-called Schur complementation. To this aim, I will present a suitable generalization of the classical known Krein - von Neumann extension.

The classical Ingham-Karamata Tauberian theorem has many applications in different fields of mathematics, varying from number theory to $C_0$-semigroup theory and is considered to be one of the most important Tauberian theorems. We will discuss how to obtain remainder estimates in the theorem if one strengthens the assumptions on the Laplace transform. Moreover, we will give new (remainder) versions of this theorem under the more general one-sided Tauberian condition of $\rho(x) \ge −f(x)$ where $f$ is an arbitrary function satisfying some regularity assumptions. The talk is based on collaborative work with Jasson Vindas.