Tim Schmatzler: The Kato square root conjecture: a second order proof

Abstract:

For any self-adjoint non-negative operator on a Hilbert space, the domain of its square root equals the form domain. In the 1960s, Tosio Kato conjectured that this equality continues to hold for sectorial second-order elliptic operators in divergence form. Until its resolution in 2001, this question played a major role in the development of new mathematical tools, in particular in harmonic analysis.
This talk gives an introduction to the square root conjecture and presents (some ideas behind) a proof for nice domains with Dirichlet boundary conditions. As opposed to earlier results on domains, the proof in this talk follows a second order approach, see [1].

[1] S. Bechtel, C. Hutchinson, T. Schmatzler, T. Tasci, M. Wittig, A second-order approach to the Kato square root problem on open sets, preprint (2024).