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