Functional model of bounded operator

The constructing of functional model for any bounded operator $T$ (contracting or not) in Hilbert space $H$ is done. It is shown that existence conditions for wave operarators $W_\pm$ within P. Lax–R. Phillips scattering scheme lead in this case to spaces $l_\beta^2$ with the weight $ \beta.$ These facts lead to Hardy spaces in the ring with the weight $W(e^{i \theta})$ which is defined by the characteristic function $S_\Delta(e^{i\theta})$ of operator $T$.