  This dissertation summarizes my investigations in operator theory during my
PhD studies.
  The first chapter is an introduction to that field of operator theory which
was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert
space operators. In the second and third chapter I characterize operators which
arise from power-bounded operators asymptotically. Chapter 4 is devoted to
provide a possible generalization of (the necessity part of) Sz.-Nagy's famous
similarity theorem. In Chapter 5 I collected my results concerning the
commutant mapping of asymptotically non-vanishing contractions. In the final
chapter the reader can find results about cyclic properties of weighted shift
operators on directed trees.
