  Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous
linear operator $\tau:D(A)\to\X$ with Range$ \tau'\cap\calH' ={0}$, $\X$ a
Banach space, we explicitly construct a family $A^\tau_\Theta$ of self-adjoint
operators such that any $A^\tau_\Theta$ coincides with the original $A$ on the
kernel of $\tau$. Such a family is obtained by giving a Kre\u\i n-like formula
where the role of the deficiency spaces is played by the dual pair $(\X,\X')$;
the parameter $\Theta$ belongs to the space of symmetric operators from $\X'$
to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of
Kiselev and Simon and so, to some extent, our results can be regarded as an
extension of it to the infinite rank case. Considering the situation in which
$\calH=L^2(\RE^n)$ and $\tau$ is the trace (restriction) operator along some
null subset, we give various applications to singular perturbations of non
necessarily elliptic pseudo-differential operators, thus unifying and extending
previously known results.
