Isometric Asymptotes of Power Bounded Operators

It is a frequently used, fruitful idea in operator theory that studying general operators, we relate them to special ones and then exploit this connection. In the present paper isometries are invoked to investigate power bounded operators. These isometries will be attached to operators in an asymptotic way and do not degenerate if the powers of the operator considered do not converge to zero in the strong operator topology. After giving the basic definitions and theorems in the first section, in Section 2 we describe how the spectra of power bounded operators relate to the spectra of their asymptotes, while in Section 3 the connection between the hyperinvariant subspace lattices is treated.