Assume that \({\mathfrak {H}}\) and \({\mathfrak {K}}\) are two real or complex Hilbert spaces, A a linear relation from \({\mathfrak {H}}\) to \({\mathfrak {K}}\), and B a linear relation from \({\mathfrak {K}}\) to \({\mathfrak {H}}\), respectively. Necessary and sufficient conditions for B to be equal to the adjoint of A are provided. Several consequences are also presented. More precisely, new characterizations for closed, skew–adjoint, selfadjoint, normal linear relations, and generalized orthogonal projections are obtained.

中文翻译：

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希尔伯特空间中线性关系的伴随

假设\（{\ mathfrak {H}} \）和\（{\ mathfrak {K}} \）是两个实数或复数Hilbert空间，甲从线性关系\（{\ mathfrak {H}} \）到\（{\ mathfrak {K}} \）和B分别是从\（{\ mathfrak {K}} \）到\（{\ mathfrak {H}} \）的线性关系。提供了使B等于A的伴随物的充要条件。还提出了一些后果。更精确地，获得了封闭，斜交，自伴，法线线性关系和广义正交投影的新特征。