# Alex Bergman: Invariant Subspaces for Second Order ODE

**Time: **
Wed 2023-10-04 11.00 - 12.00

**Location: **
Albano, Cramérrummet

**Participating: **
Alex Bergman (Lund)

**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 *L*^{2}(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.