Spectral Aspects of a Class of Differential Operators

Abstract

Recently Gisli Masson and Boris Shapiro initiated the study of a class of differential operators defined as follows: for each monic polynomial Q, let T Q be the operator mapping f to (d/dz) k f,where k = deg Q and f belongs, say, to the set E of all entire functions of a complex variable. They showed (Q being now fixed) that for each non-negative integer m there is a unique monic polynomial f m of degree m which is an eigenfunction of T Q. Moreover the corresponding eigenvalue is positive, and depends only on m and k,but not on the specific choice of Q with degree k. They studied the location of the zeros of f m . The goal of this paper is to study natural spectral questions arising from their findings. Our main results are: 1) In the space E there are no eigenfunctions of T Q other than the {f m }. 2) In a certain (explicitly given) Hilbert space H of entire functions all T Q with deg Q = k are similar to one and the same self-adjoint positive operator. (This “explains” both why the eigenvalues are the same for all Q of given degree, and why they are real and positive). A closely related result is that { f n,} do not merely span H, but are a Riesz basis for this space. These results are proved using standard tools of perturbation theory. In a concluding section attention is drawn to an operator in a sense dual to T Q, whose eigenfunctions (which now are entire, but not in general polynomials) relate in interesting ways to { f n}.