Current work defines Schur representation of a bilinear operator \(T: H \times H \rightarrow H\), where H is a separable Hilbert space. Introducing the concepts of self-adjoint bilinear operators, ordered eigenvalues and eigenvectors, we prove that if T is compact, self-adjoint, and its eigenvalues are ordered, then T has a Schur representation, thus obtaining a spectral theorem for T on real Hilbert spaces. We prove that the hypothesis of the existence of ordered eigenvalues is fundamental.

中文翻译：

---

Hilbert空间中双线性紧算子的谱定理

当前工作定义了双线性算子（T：H \ times H \ rightarrow H \）的Schur表示，其中H是可分离的希尔伯特空间。引入自伴随双线性算子，有序特征值和特征向量的概念，我们证明了如果T是紧实的，自伴随的并且其特征值是有序的，则T具有Schur表示，从而获得实Hilbert上T的谱定理空格。我们证明了有序特征值存在的假设是基本的。