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.

给定收缩甲上的Hilbert空间\（{\ mathcal {H}} \） ，操作者Ť上\（{\ mathcal {H}} \）被认为是甲-invariant如果\（\ langle的Tx中，x \ rangle = \ langle TAx，Ax \ rangle \）对于每个\（x \ in {\ mathcal {H}} \}），使得\（\ Vert Ax \ Vert = \ Vert x \ Vert \）。在A的两个缺陷指数都等于1的特殊情况下，我们表明每个A不变算子都是对无穷大线性变换的\（{\ mathcal {H}} \\}的压缩，该线性变换用最小unit A的膨胀。Sarason在其他假设下证明了这一结果，即A属于\（C_ {00} \）类，从而导致截断的Toeplitz算符的固有特征。我们还根据截断的Toeplitz运算符的其他结果来适应更一般的上下文。