Multiplicative perturbations of stable and convergent operators

Some familiar classes of stable Hilbert-space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie. Analogous, but less familiar, classes of convergent operators are examined with the same aim. The classes considered are often sets of products A where is a given set of diagonal or Hermitian matrices and A is a single matrix. The Aʹs for which A is a set of stable or convergent operators are sometimes characterized.