The Kreĭn-Langer problem for Hilbert space operator valued functions on the band (Q1304742)

The authors consider functions \(f:(-2a, 2a)\times \Gamma\to{\mathcal L}(H)\), where \(a>0\), \(\Gamma\) is a topological Abelian group, and \({\mathcal L}(H)\) is the algebra of continuous linear operators on the Hilbert space \(H\). Such a function is \(k\)-indefinite if: 1) \(f(x- y)= f(y- x)^*\) for all \(x\) and \(y\) in the band \(B= (-a,a)\times \Gamma\), and 2) the Hermitian matrix \((\langle f(x_j- x_k)h_j, h_k\rangle_H)^n_{j,k= 1}\) has at most \(k\) negative eigenvalues, but exactly \(k\) for some choice of \(n\in \mathbb{N}\), \(x_1,\dots, x_n\in(-a, a)\times\Gamma\) and \(h_1,\dots, h_n\in H\). In the case of positively defined functions, when \(k=0\), the problem of existence, uniqueness and parametrization of an extension is already solved. Based on results concerning the associated strongly continuous local semigroup of isometries, obtained by the authors in previous works, they solve the similar Kreĭn-Langer problem for \(k\neq 0\).