We describe a recent result of M. Hager, stating roughly that for nonselfadjoint ordinary differential operators with a small random perturbation we have a Weyl law for the distribution of eigenvalues with a probability very close to 1.

We investigate selfadjoint C0-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the L∞ operator norm of the semigroup. As an application we prove a new L∞-bound for the torsion function of a Euclidean domain that is cl...

In this note we establish the large time limit non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.

We consider the system of homogeneous Dirichlet boundary value problems (*) Liu = A[an(x)u + a\2(x)v], L2V = ti[ai2{x)u + a.22{x)v] in a smooth bounded domain fi C R", where Li and £2 are formally selfadjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detai...

The paper continues the study of differential Banach *algebras AS and FS of operators associated with symmetric operators S on Hilbert spaces H. The algebra AS is the domain of the largest *-derivation δS of B(H) implemented by S and the algebra FS is the closure of the set of all finite rank operators in AS with respect to the norm ‖A‖ = ‖A‖+‖δS(A)‖. When S is selfadjoint, FS is the domain of ...

We study the essential self-adjointness for real principal type differential operators. Unlike elliptic case, we need geometric conditions even operators on Euclidean space with asymptotically constant coefficients, and prove under null non-trapping condition.