Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is consider...

A composition operator Tbf = f o b is completely continuous on H1 if and only if \b\ < 1 a.e. If the adjoint operator Tg is completely continuous on VMOA , then 7¿ is completely continuous on Hl . Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of ce...

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

If Ω is a ball in Rn (n ≥ 2), then the boundary integral operator of the double layer potential for the Laplacian is self-adjoint on L2(∂Ω). In this paper we prove that the ball is the only bounded Lipschitz domain on which the integral operator is self-adjoint.

In this paper, we prove that the Hodge-Laplace operator on strongly pseudoconvex compact complex Finsler manifolds is a self-adjoint elliptic operator. Then, from the decomposition theorem for self-adjoint elliptic operators, we obtain a Hodge decomposition theorem on strongly pseudoconvex compact complex Finsler manifolds. M.S.C. 2010: 53C56, 32Q99.

Given a causal analytic nonlinear input-output system represented as a Chen-Fliess functional series, this paper investigates how to apply an existing notion of a nonlinear Hilbert adjoint operator to explicitly compute a corresponding adjoint operator. The method is demonstrated for the bilinear case.

Given a self-adjoint operator H0, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H0, F) ⊂ R is defined, such that for all points of this set the wave and the scattering matrices can be defined unambiguously. Many well-known properties of the wave and sc...

We describe how self-adjoint ordered operator spaces, also called non-unital systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. explain this perspective has several advantages over other notions of literature. In particular, category includes injective objects and Webster-Winkler-type duality theorem, both which we show generally fail syste...