36 research outputs found
Schur functions and their realizations in the slice hyperholomorphic setting
we start the study of Schur analysis in the quaternionic setting using the
theory of slice hyperholomorphic functions. The novelty of our approach is that
slice hyperholomorphic functions allows to write realizations in terms of a
suitable resolvent, the so called S-resolvent operator and to extend several
results that hold in the complex case to the quaternionic case. We discuss
reproducing kernels, positive definite functions in this setting and we show
how they can be obtained in our setting using the extension operator and the
slice regular product. We define Schur multipliers, and find their co-isometric
realization in terms of the associated de Branges-Rovnyak space
Around the Van Daele–Schmüdgen Theorem
For a {bounded} non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show how the existence of such subspaces leads to various {pathological} properties of {unbounded} self-adjoint operators related to von Neumann theorems \cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van Daele-Schm\"udgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to refine them. We also develop {a new systematic approach, which allows to construct for any {unbounded} densely defined symmetric/self-adjoint operator T infinitely many pairs of its closed densely defined restrictions T_k\subset T such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and \dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T