Invariant subspaces of the shift operator. Axiomatic approach

There is axiomatically described the class of spaces $Y$ (resp. $X$) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator $f(z)\mapsto zf(z)$ (resp. the inverse shift $f(z)\mapsto z^{-1}(f(z)-f(0))$) are constructed just like the Hardy space $H^2$. It is proved that as $X$ one can take, for example, the space $H^1$, the disk-algebra $C_A$, the space $U_A$ of all uniformly convergent power series; and as $Y$ the space of integrals of Cauchy type $L^1/H^1_-$, the space $VMO_A$. There is also obtained an analog for the space $U_A$ of W. Rudin's theorem on $z$-invariant subspaces of the space $C_A$.