Alex Bergman: Invariant Subspaces for Second Order ODE

Abstract.
In 1939 Gelfand posed the problem of determining the invariant subspaces of the Volterra operator,

\(V f(x) = \int_0^x f(t)dt, \)

on L²(0, 1) (a closed subspace M is called invariant if \(V M \subset M\)). This attractive problem has been solved by numerous authors. Crucial in certain approaches is that V is the right inverse of the differential operator \(D = d/dx\). More recently Aleman, Baranov, Belov, and Korenblum considered subspaces of \(C^\infty\) invariant under D. The papers [2, 1, 3] give a complete description when the spectrum is discrete. In this talk we consider generalizations of these result to pairs of operators L and T, such that \(LT = I\) and for which L which has self-adjoint restrictions. In particular we have in mind operators L and T arising from second order ODE, e.g., Sturm–Liouville equations or more generally canonical systems. This is based on recent joint work with Alexandru Aleman.

References
[1] A. Aleman, A. Baranov, and Y. Belov. Subspaces of \(C^\infty\) invariant under the differentiation. J. Funct. Anal., 268(8):2421–2439, 2015.
[2] A. Aleman and B. Korenblum. Derivation-invariant subspaces of \(C^\infty\). Comput. Methods Funct. Theory, 8(1-2):493–512, 2008.
[3] A. Baranov and Y. Belov. Synthesizable differentiation-invariant subspaces. Geom. Funct. Anal., 29(1):44–71, 2019.