Given Hilbert space operators A, B, C, one seeks an operator D such that $\| \left( \begin{array}{cc} A ~ C\\ B ~ D\end{array} \right) \| \leqq \mu$ for a pre-assigned bound $\mu$. In some contexts only the existence of D is called for; in others, one may want to know something about those D which work. This paper gives an exposition of the main theorems and of two of the many applications.

We consider the vibration problem for an elastic bounded body with small compressibility, which is associated with a small parameter $\epsilon$, As $\epsilon \downarrow 0$ this is a stiff perturbation problem with non-analytic character, (in particular, the domain of the operator for $\epsilon = 0$ is not dense in a standard space, rdhereas for $\epsilon \neq 0$ it is). Nevertheless, analytic perturbation theory applies and we prove that the solution corresponding to each point of the resolvent set of the $\epsilon = 0$ problem may be expanded as a series convergent: for small $| \epsilon |$ moreover, eigenvalues and eigenvectors have holomorphic expansions for small $| \epsilon |$. Explicit computation of the first terms of the perturbation is given. The asymptotic behaviour of eigenvalues for large values of the spectral parameter is also given, and we show that it is not holomorphic in $ \epsilon $. The preceding techniques are applied to the problem of vibrations of a slightly viscous compressible fluid in a bounded vessel; an implicit function argument allows us to prove that infinitely many real eigenvalues converge as $ \epsilon + 0$ in an analytic way to the origin which is an eigenvalue of infinite multiplicity of the problem for $ \epsilon = 0$.