Orthogonal and skew-symmetric operators in real Hilbert space
As the authors comment at the end of the first section of the paper under review, the ``real Hilbert spaces have been treated rather stepmotherly''. With this motivation, they prove a well known theorem in the case of complex Hilbert spaces for the case of a real Hilbert space. More precisely, they show that, if \(H\) is a real Hilbert space, \(\mathcal{U}(H)\) is an ideal in \(\mathcal{L}(H)\) and \(\tau\) is a linear form on \(H\), then the following statements are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] \(\tau\) is a trace on \(\mathcal{U}(H)\); \item[(ii)] \(\tau(U^*TU)= \tau(T)\) for all \(T \in \mathcal{U}(H)\) and all orthogonal operators \(U \in \mathcal{L}(H)\); \item[(iii)] \(\tau(U^*TU)= \tau(T)\) for all symmetric operators \(T \in \mathcal{U}(H)\) and all orthogonal operators \(U \in \mathcal{L}(H)\) and \(\tau(S)=0\) for all skew-symmetric operators \(S \in \mathcal{U}(H)\). \end{itemize}} The proof of this important theorem is obtained by means of two main results established in the paper: \((a)\) Every operator on a real Hilbert space is a finite linear combination of orthogonal operators; \((b)\) every trace on \(\mathcal{U}(H)\) vanishes on the skew-symmetric operators. In addition to these very interesting results, a complement is presented in the last section to discuss the ``relationship'' between theorems like the one above, which appear in the context of operator ideals, and theorems of the same type that appear in the context of operator algebras such as \(C^*\)-algebras.
scientific article; zbMATH DE number 6153947
scientific article