  Recent results concerning the linear dynamics and mean ergodicity of compact
operators in Banach spaces, together with additional new results, are employed
to investigate various spectral properties of generalized Ces\`aro operators
acting in large classes of classical BK-sequence spaces. Of particular interest
is to determine the eigenvalues and the corresponding eigenvectors of such
operators and to decide whether (or not) the operators are power bounded, mean
ergodic and supercyclic.
