We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension.

中文翻译：

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实线上实解析函数空间上托普利兹算子的单边可逆性

我们证明了实线上实解析函数空间上的 Toeplitz 算子是左可逆的当且仅当它是一个单射 Fredholm 算子，它是右可逆的当且仅当它是一个满射 Fredholm 算子。特征是根据运算符符号的绕数给出的。我们的结果意味着 Toeplitz 算子（以及它的伴随算子）的范围是互补的，当且仅当它是有限余维的。类似地，当且仅当它是有限维的，托普利兹算子（以及它的伴随）的核是互补的。