We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

We discuss the three-space problem on discreteness for the Jacobson topology on the spectrum of a C*-algebra in detail. As an application, it is shown that a C*-algebra A is a dual C*-algebra if and only if a closed ideal I of A and the quotient A/I are dual C*-algebras and the open central projection, in the second dual of A, corresponding to I is a multiplier for A.

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane C+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

We compute critical groups in semilinear elliptic boundary-value problems in which the nonlinear term may fail to have asymptotic limits at zero and at infinity. As applications, we prove several new existence results.