LAUSR

Abstract Cuckovic and Paudyal characterized the lattice of invariant subspaces of the operator T in the Hardy–Hilbert space H 2 ( D ) , where they studied the special case (when p = 2 ) of the space S p ( 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 H p ( D ) if and only if T z ( M ) is an invariant subspace of M z on S 0 p ( D ) , if and only if T z ( M ) is a closed ideal of S 0 p ( D ) . Furthermore, we provide certain Beurling-type invariant subspaces of M z on S p ( D ) and S 0 p ( D ) . Then, we investigate the boundedness of the operators T g and I g on S p ( D ) . Finally, we investigate the spectrum of multiplication operator M g on S p ( D ) , the isometric multiplication operators and the isometric zero-divisors on S p ( D ) .