Isometric embeddings of the spaces \(K_{\Theta}\) into spaces of the upper half-plane (Q2773586)

The author studies a class of positive measures \(\mu\) defined on \(\mathbb R\) such that NEWLINE\[NEWLINE \int_{\mathbb R}|f|^2d\mu = \|f\|^2\quad \text{for all~} f\in K_{\Theta} = H^2 \ominus \Theta H^2, NEWLINE\]NEWLINE where \(\Theta\) is the so-called inner function in the upper half-plane of \(\mathbb C\). In other words, the author studies measures \(\mu\) which guarantee an isometric inclusion of spaces \(K_{\Theta} \subset L^2(\mu)\). This is a primary problem of the article.NEWLINENEWLINENEWLINEThe main aim is to expose a connection between the results about isometric inclusions of the de Branges spaces and isometric inclusion theorems for co-invariance subspaces of the translation operator in \(\mathbb C^+\). The author proves that, in spite of the fact that those results are similar, there exists an essential difference between them which is caused by the passage from the (compact) unit circle to the real line.