We consider a one parameter family of Laplacians on a closed manifold and study the semi-classical limit of its analytically parametrized eigenvalues. Our results establish a vector valued analogue of a theorem for scalar Schrödinger operators on Euclidean space by Luc Hillairet which applies to geometric operators like Witten's Laplacian on differential forms.

We characterize sampling and interpolating sets with derivatives in weighted Fock spaces on the complex plane in terms of their weighted Beurling densities.

We introduce the new concepts of pseudo numerical range for operator functions and families of sesquilinear forms as well as the pseudo block numerical range for operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order 0 and not even . As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.

This paper is devoted to the analysis of the single layer boundary integral operator for the Dirac equation in the two- and three-dimensional situation. The map is the strongly singular integral operator having the integral kernel of the resolvent of the free Dirac operator and belongs to the resolvent set of . In the case of smooth boundaries fine mapping properties and a decomposition of in a 'positive' and 'negative' part are analyzed. The obtained results can be applied in the treatment of Dirac operators with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions that are combined in a critical way.

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols and , we show that is compact if and only if vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of on the whole polydisc might not imply the compactness of . On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

We give new characterizations of the optimal data space for the -Neumann boundary value problem for the operator associated to a bounded, Lipschitz domain . We show that the solution space is embedded (as a Banach space) in the Dirichlet space and that for , the solution space is a reproducing kernel Hilbert space.