The main result of this paper shows that the existence of commuting normal extension (c.n.e.) for an arbitrary family of commuting subnormal operators can be determined by considering appropriate families of multivariable weighted shifts. In proving this some known criteria for c.n.e. are generalized. It is also shown that a family of jointly quasi-normal operators has c.n.e.

An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given. An integral representation for their eigenvalues is obtained. This leads to estimates of the differences in eigenvalues and the number of points of degeneracy.

Let ˜ be an equivalence relation on a topological space X. A point x ε X i s stable with respect to ˜ if it is in the interior of an equivalence class. We may also add, if ambiguity arises, that x is stable under perturbations in X. Let E be a Banach space, and let L(E) be the Banach space of continuous linear endomorphisms of E, with norm given by |T| = sup{ |T(x) | : |x| = 1}. In this paper we discuss stability of elements of L(E) with respect to some natural equivalence relations.