Volterra type operators on [formula omitted] spaces

Abstract

Čučković and Paudyal characterized the lattice of invariant subspaces of the operator T in the Hardy–Hilbert space H2(D), where they studied the special case (when p=2) of the space Sp(D). We generalize some of their works to the general case when 1≤p<∞ and determine that M is an invariant subspace of T on Hp(D) if and only if Tz(M) is an invariant subspace of Mz on S0p(D), if and only if Tz(M) is a closed ideal of S0p(D). Furthermore, we provide certain Beurling-type invariant subspaces of Mz on Sp(D) and S0p(D). Then, we investigate the boundedness of the operators Tg and Ig on Sp(D). Finally, we investigate the spectrum of multiplication operator Mg on Sp(D), the isometric multiplication operators and the isometric zero-divisors on Sp(D).