A complex scalar ? is said to be an extended eigenvalue of a bounded linear operator on Hilbert space if there nonzero X such that AX=?XA. The results in this paper provide full solution the problem computing eigenvalues for those composition operators C? induced Hardy H2(D) by fractional transformations ? unit disk.

In this paper we study the boundedness of composition operators on weighted Bergman spaces and Hardy space over polydisc Dn. Studying volume sublevel sets show for which n necessary conditions obtained by Bayart are sufficient. For arbitrary prove rank sufficiency theorem which, in particular, provides us with a simple criterion describing bidisc. Such consistent characterization is classical t...