Past Functional Analysis Seminar

30 October 2012
17:00
to
18:23
Vesselin Petkov
Abstract
We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f,\: t \geq 0$. The solutions $u = e^{tG_b}f$ with eigenfunctions $f$ of the generator $G_b$ with eigenvalues $\lambda,\: \Re \lambda < 0,$ are called asymptotically disappearing (ADS). If we have (ADS), the wave operators are not complete and the inverse back-scattering problems become complicated. We examine the spectrum of the generator $G_b$ and we show that this spectrum in the open half plane $\Re \lambda < 0$ is formed by isolated eigenvalues with finite multiplicity. The existence of (ADS) is a difficult problem. We establish the existence of (ADS) for the Maxwell system in the exterior of a sphere. We will discuss some other applications related to the existence of (ADS).
  • Functional Analysis Seminar
23 October 2012
17:00
to
18:23
Vladimir Muller
Abstract
Let $T_1,\dots,T_n$ be bounded linear operators on a complex Hilbert space $H$. We study the question whether it is possible to find a unit vector $x\in H$ such that $|\langle T_jx, x\rangle|$ is large for all $j$. Thus we are looking for a generalization of the well-known fact for $n = 1$ that the numerical radius $w(T)$ of a single operator T satisfies $w(T)\ge \|T\|/2$.
  • Functional Analysis Seminar
15 May 2012
17:00
to
18:10
Fritz Gesztesy
Abstract
We extend the classical trace formula connecting the trace of resolvent dif- ferences of two (not necessarily self-adjoint) operators A and A0 with the logarithmic derivative of the associated perturbation determinant from the standard case, where A and A0 have comparable domains (i.e., one contains the other) to the case where their square root domains are comparable. This is done for a class of positive-type operators A, A0. We then prove an abstract result that permits to compare square root domains and apply this to the concrete case of 2nd order elliptic partial di erential operators in divergence form on bounded Lipschitz domains. This is based on various joint work with S. Hofmann, R. Nichols, and M. Zinchenko.
  • Functional Analysis Seminar
15 May 2012
09:30
to
10:45
Abstract
The stochastic Weiss conjecture is the statement that for linear stochastic evolution equations governed by a linear operator $A$ and driven by a Brownian motion, a necessary and sufficient condition for the existence of an invariant measure can be given in terms of the operators $\sqrt{\lambda}(\lambda-A)^{-1}$. Such a condition is presented in the special case where $-A$ admits a bounded $H^\infty$-calculus of angle less than $\pi/2$. This is joint work with Jamil Abreu and Bernhard Haak.
  • Functional Analysis Seminar
24 April 2012
17:00
to
18:33
Nicholas Young
Abstract
A theorem of R. Nevanlinna from 1922 characterizes the Cauchy transforms of finite positive measures on the real line as the functions in the Pick class that satisfy a certain growth condition on the real axis; this result is important in the spectral theory of self-adjoint operators. (The Pick class is the set of analytic functions in the upper half-plane $\Pi$ with non-negative imaginary part). I will describe a higher-dimensional analogue of Nevanlinna's theorem. The $n$-variable Pick class is defined to be the set of analytic functions on the polyhalfplane $\Pi^n$ with non-negative imaginary part; we obtain four different representation formulae for functions in the $n$-variable Pick class in terms of the ``structured resolvent" of a densely defined self-adjoint operator. Structured resolvents are analytic operator-valued functions on the polyhalfplane with properties analogous to those of the familiar resolvent of a self-adjoint operator. The types of representation that a function admits are determined by the growth of the function on the imaginary polyaxis $(i\R)^n$.
  • Functional Analysis Seminar

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