Abstract.
On the Hardy space over the bidisk H2(D2), the Toeplitz operators \(T_{{z_{1} }}\)and \(T_{{z_{2} }} \) are unilateral shifts of infinite multiplicity. A closed subspace M is called a submodule if it is invariant for both \(T_{{z_{1} }} \) and \(T_{{z_{2} }} \) . The two variable Jordan block (S1, S2) is the compression of the pair \(T_{{z_{1} }}, T_{{z_{2} }}\) to the quotient H2(D2) ⊖M. This paper defines and studies its defect operators. A number of examples are given, and the Hilbert-Schmidtness is proved with good generality. Applications include an extension of a Douglas-Foias uniqueness theorem to general domains, and a study of the essential Taylor spectrum of the pair (S1, S2). The paper also estabishes a clean numerical estimate for the commutator [S1*, S2] by some spectral data of S1 or S2. The newly-discovered core operator plays a key role in this study.
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Yang, R. On Two Variable Jordan Block (II). Integr. equ. oper. theory 56, 431–449 (2006). https://doi.org/10.1007/s00020-006-1422-8
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DOI: https://doi.org/10.1007/s00020-006-1422-8