Let H 2 (D 2 ) be the Hardy space over the bidisk. For sequences of Blaschke products {ϕ n (z): −∞ < n < ∞} and {ψ n (w): −∞ < n < ∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H 2 (D 2 ) M for the pair of operators T * z and T * w .
Let H 2 ðD 2 Þ be the Hardy space over the bidisk. Let fj n ðzÞg nf0 be a sequence of one variable inner functions such that j n ðzÞ=j nþ1 ðzÞ is a nonconstant inner function for every n f 0. Associated with them, we have an invariant subspace M of H 2 ðD 2 Þ. When j 0 ðzÞ is a Blaschke product, it is determined rankðM m wMÞ for the fringe operator F z on M m wM and rank M as an invariant subspace of H 2 ðD 2 Þ.
An elementary proof of the Aleman, Richter and Sundberg theorem concerning the invariant subspaces of the Bergman space is given. (2000). Primary 47A15, 32A35; Secondary 47B35.
Mathematics Subject Classification
Linear sums of two composition operators of the multi-dimensional Fock space are studied. We show that such an operator is bounded only when both composition operators in the sum are bounded. So, cancelation phenomenon is not possible on the Fock space, in contrast to what have been known on other well-known function spaces over the unit disk. We also show the analogues for compactness and for membership in the Schatten classes. For linear sums of more than two composition operators the investigation is left open.
Let B be the Bergman shift on the Bergman space L 2 a over the open unit disk and let I be a nontrivial invariant subspace of L 2 a . Let PI be the orthogonal projection from L 2 a onto I. It is proved that PI B(L 2 a I) is not dense in I if and only if I ∩ D = {0}, where D is the Dirichlet space. It is also discussed some related topics. Mathematics Subject Classification (2010). Primary 47A15; Secondary 32A35 47B35.
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