Talks

Given an elliptic operator L=-divA∇ on L^2(R^n) one has by the Kato square root property that dom L^½=H^1(R^n) and

c||∇f||_2 ≤ ||L^½f||_2 ≤ C||∇f||_2

for some c,C>0 and f∈H^1(R^n). The right inequality above beeing equivalent to the Riesz transform ∇L^(-½) being bounded on L^2(R^n). The subject of the talk will be for what other p-norms the above inequality, specifically boundedness of the Riesz transform holds. It turns out that they are essentially the same p as for which the semigroup generated by -L is bounded on L^p(R^n).

Zoomlink:
https://tuhh.zoom.us/j/83250070589?pwd=UZ52e5cgqaexuY9SnleyZo01MmukYW.1

In this talk, we will discuss a very recent operator-theoretic result concerning second-order elliptic equations that yields a compactness theorem in the homogenization theory of the biharmonic equation, i.e., a fourth-order problem.

Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann’s Cayley transform. Using ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators. This is joint work with K. Landsman (Nijmegen, Netherlands).

Zoomlink:
https://tuhh.zoom.us/j/81617046707?pwd=tTkia9CCoCuaiEaEsrCmTqlMr1cfOT.1

Suppose that A, B, and C are compact selfadjoint operators on a Hilbert space such that A+B+C=0. We show that the possible eigenvalues of these operators are described by certain combinatorial objects called hives. We will also see that the Horn inequalities (first described in the compact case in joint work with Li and Timotin) can also be obtained from this hive model.