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

Reflexive linear transformations

AbstractA linear operator A is called reflexive if the only operators that leave invariant the invariant subspaces of A are the operators in the weak closure of the algebra of polynomials in A. In this note we completely characterize reflexive operators on finite-dimensional spaces

The hyperinvariant subspace lattice of a linear transformation

AbstractA description of the lattice of hyperinvariant subspaces of a linear transformation on a finite-dimensional vector space is given. Various properties of such lattices are determined, as well as implications between linear-algebraic and lattice-theoretic properties

Intersections of nest algebras in finite dimensions

AbstractIf M,N are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dimH. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dimH. For any two maximal nests M,N there exists a basis {f1,f2,…,fn} of H and a permutation π such that M={(0)}∪{Mi:1⩽i⩽n} and N={(0)}∪{Ni:1⩽i⩽n}, where Mi= span{f1,f2,…,fi} and Ni= span{fπ(1),fπ(2),…,fπ(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dimH, precisely when π(j)=n−j+1,1⩽j⩽n. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras