On the proof of Beurlingʹs theorem on z- invariant subspaces

We give an other proof of the classical Beurling theorem on z -invariant subspaces in the classical Hardy space over the open unit disk , in which the concept of Berezin symbol is not used (see M.T. Karaev [On some problems related to Berezin symbols, C. R. Acad. Sci. Paris, Ser. I 340 (10) (2005) 715–718]). This improves the authorʹs proof in Karaev (2005).