Abstract
Given a contraction A on a Hilbert space \({\mathcal {H}}\), an operator T on \({\mathcal {H}}\) is said to be A-invariant if \(\langle Tx,x\rangle =\langle TAx,Ax\rangle \) for every \(x\in {\mathcal {H}}\) such that \(\Vert Ax\Vert =\Vert x\Vert \). In the special case in which both defect indices of A are equal to 1, we show that every A-invariant operator is the compression to \({\mathcal {H}}\) of an unbounded linear transformation that commutes with the minimal unitary dilation of A. This result was proved by Sarason under the additional hypothesis that A is of class \(C_{00}\), leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other results about truncated Toeplitz operators.
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Bercovici, H., Timotin, D. Operators invariant relative to a completely nonunitary contraction. Math. Z. 299, 1631–1649 (2021). https://doi.org/10.1007/s00209-021-02731-9
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DOI: https://doi.org/10.1007/s00209-021-02731-9