Local semigroups of isometries in \(\Pi_ \kappa\)-spaces and related continuation problems for \(\kappa\)-indefinite Toeplitz kernels
Let \(I=\mathbb{R}\) or \(I=[-a,a]\) with \(0<a<\infty\). An Hermitian kernel \(F(x,y)\) on \(I\) is said to be \(\kappa\)-indefinite with \(\kappa<\infty\) if for any \(n\) and any points \(x_ 1,\dots,x_ n\) in \(I\) the \(n\times n\) matrix \([F(x_ i,x_ j)]\) has at most \(\kappa\) negative eigenvalues, and exactly \(\kappa\) for some choice. \(F\) is called a Toeplitz kernel if \(F(x,y)=f(x-y)\) for some continuous function \(f\) on the closure of \(I-I\), while it is called a generalized Toeplitz kernel (G.T.K.) if there are four continuous functions \(f_{\alpha\beta}\) (\(\alpha,\beta=1,2\)) on the closure of \(I_ \alpha-I_ \beta\) such that \(F(x,y)=f_{\alpha\beta}(x-y)\) for \(x\in I_ \alpha\), \(y\in I_ \beta\), where \(I_ 1=I\cap[0,+\infty)\) and \(I_ 2=I\cap(- \infty,0]\). \textit{M. G. Krein} and \textit{H. Langer} [J. Operator Theory 13, 299-417 (1985; Zbl 0608.47038)] proved that every \(\kappa\)-indefinite Toeplitz kernel on \(I=[-a,a]\) can be extended to a \(\kappa\)-indefinite Toeplitz kernel on \(\mathbb{R}\). In the present paper the authors generalize the Krein-Langer theorem to the case of \(\kappa\)-indefinite G.T.K. Considering a local semigroup of isometries on a Pontryagin space with generator \(F\) they reduce the problem to the assertion that every strongly continuous local semigroup of isometries can be extended to a unitary group on larger Pontryagin space with same index.
strongly continuous local semigroup of isometries
scientific article; zbMATH DE number 92245
scientific article