Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy-Foias theorem for two pairs of dilations
The authors prove the following transference result: Let \(H_ 1\) and \(H_ 2\) be two Hilbert spaces, \(U_ 1\in L(H_ 1)\) and \(U_ 2\in(H_ 2)\), two unitary operators, and \(N_ 1\subset N_ 2\) and \(N_ 2\subset H_ 2\), two invariant subspaces, i.e., \[ U_ 1N_ 1\subset N_ 1,\quad U_ 2N_ 2\subset N_ 2, \] so that \(T_ 1=U_ 1|_{N_ 1}\) and \(T_ 2=U_ 2|_{N_ 2}\) are semiunitary operators in \(N_ 1\) and \(N_ 2\), respectively. Given a unitary coupling \(T\in L(N)\), of \(T_ 1\) and \(T_ 2\), there exists a unitary coupling \(U\in L(H)\) of \(U_ 1\) and \(U_ 2\) such that \[ \langle f_ 1,f_ 2\rangle_ H=\langle f_ 1,f_ 2\rangle_ N,\quad\forall f_ 1\in N_ 1,\;f_ 2\in N_ 2. \] This result allows them to prove a lifting theorem for invariant forms acting in scattering systems with one or two evolution groups.
scattering systems with one or two evolution groups
scientific article; zbMATH DE number 152051