Curvature and discrete series representation of \(SL_ 2({\mathbb{R}})\)
Let \({\mathbb{D}}\) be the open unit disk in the complex plane. A bounded linear operator T on a Hilbert space H is of class \(B_ 1({\mathbb{D}})\) if T-\(\zeta\) I is surjective, dim ker (T-\(\zeta\) I)\(=1\) for \(\zeta\in D\) and \(\bigvee_{\zeta \in {\mathbb{D}}}\ker (T-\zeta I)\) is dense in H. This class of operators was first studied by \textit{M. J. Cowen} and \textit{R. G. Douglas} [Acta Math. 141, 187-261 (1978; Zbl 0427.47016)] as was their associated curvature \(k_ T(\zeta).\) In this paper, the author first gives a straightforward proof of their result that two operators in \(B_ 1({\mathbb{D}})\) are unitarily equivalent if and only if their curvatures are equal. Using this, the author proves a previous theorem of him which characterizes those operators T in \(B_ 1({\mathbb{D}})\) which are unitarily equivalent to \(\phi_ s(T)\) for all Möbius transforms \(\phi_ s\) of \({\mathbb{D}}\). This latter result is then used to give a new proof of the unitary representation of the group \(SL_ 2({\mathbb{R}})\) via an operator on some Hilbert space of analytic functions on \({\mathbb{D}}\).
unitary representation of the group \(SL_ 2({\mathbb{R}})\)
scientific article; zbMATH DE number 3962536
scientific article