OTTER
Operator Theory Talks for Early Researchers

The monthly problem

One problem a month, pitched at early-graduate level — posted here and in OTTER News.

October 2026 · Problem 1 · sample

Let \(S\) be the unilateral shift on the Hardy space \(H^2\), that is, \((Sf)(z) = z f(z)\).

  1. Show that \(S\) has no eigenvalues.
  2. Show that every \(\lambda\) in the open unit disk \(\mathbb{D}\) is an eigenvalue of \(S^*\), with eigenvector the Szegő kernel $$ k_\lambda(z) = \frac{1}{1 - \overline{\lambda}\, z}. $$
  3. Conclude that the spectrum of \(S\) is the closed unit disk \(\overline{\mathbb{D}}\).

Send your solution to ottermathtalks@gmail.com by October 31, 2026. Favorite solutions are featured — with your name — in the next issue of OTTER News.

The problem archive begins with the October 2026 relaunch.