There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z^-1(f(z)-f (0))) are constructed just like the Hardy space H². It is proved that as Χ one can take, for example, the space H¹, the disk-algebra C_A, the space U_A of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L¹/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.