Abstract
We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that Tzn is similar to the sum of n copies of the Bergman shift.
Highlights
Let T be a bounded linear operator on a Hilbert space H;a subspace M of H is called an invariant subspace of T ifT(M) ⊆ M and a reducing subspace of T if M is an invariant subspace of T and T∗
The Toeplitz operator Tφ on L2a(D) with symbol φ ∈ L∞(dA(z)) is defined by (Tφf)(z) = P(φf)(z); it is called an analytic Toeplitz operator if φ ∈ H∞(D)
Aleman et al defined and studied the Hardy space of a slit domain and in particular they studied the invariant subspace of the slit disk; one can consult [17] for details
Summary
T(M) ⊆ M and a reducing subspace of T if M is an invariant subspace of T and T∗. A reducing subspace M of T is called minimal if for every reducing subspace Mof T such that M ⊆ M either M = M or M = 0. Thomson [1, 2] showed that it suffices to study reducing subspace of TB for a finite Blaschke product in the case of Hardy space. Zhu studied the reducing subspaces of TB for a Blaschke product of order 2 firstly and showed that TB has exactly two distinct minimal reducing subspaces (cf [3]). Guo et al showed that in general this is not true (cf [4]), and they found that the number of minimal reducing subspaces of TB equals the number of connected components of the Riemann surface of B(z) = B(w) when the order of B is 3, 4, 6 They conjectured that the number of minimal reducing subspaces of TB equals the number of connected components of the Riemann surface of B(z) = B(w) for any finite Blaschke product (called the refined Zhu’s conjecture, cf [4]). We analyze the concrete examples to see what are the possible models for these restrictions for further research
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