357 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
On the class SI of J-contractive functions intertwining solutions of linear differential equations
In the PhD thesis of the second author under the supervision of the third
author was defined the class SI of J-contractive functions, depending on a
parameter and arising as transfer functions of overdetermined conservative 2D
systems invariant in one direction. In this paper we extend and solve in the
class SI, a number of problems originally set for the class SC of functions
contractive in the open right-half plane, and unitary on the imaginary line
with respect to some preassigned signature matrix J. The problems we consider
include the Schur algorithm, the partial realization problem and the
Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence
between elements in a given subclass of SI and elements in SC. Another
important tool in the arguments is a new result pertaining to the classical
tangential Schur algorithm.Comment: 46 page
Generalized Fock spaces and the Stirling numbers
The Bargmann-Fock-Segal space plays an important role in mathematical
physics, and has been extended into a number of directions. In the present
paper we imbed this space into a Gelfand triple. The spaces forming the
Fr\'echet part (i.e. the space of test functions) of the triple are
characterized both in a geometric way and in terms of the adjoint of
multiplication by the complex variable, using the Stirling numbers of the
second kind. The dual of the space of test functions has a topological algebra
structure, of the kind introduced and studied by the first named author and G.
Salomon.Comment: revised versio
Canonical, squeezed and fermionic coherent states in a right quaternionic Hilbert space with a left multiplication on it
Using a left multiplication defined on a right quaternionic Hilbert space, we
shall demonstrate that various classes of coherent states such as the canonical
coherent states, pure squeezed states, fermionic coherent states can be defined
with all the desired properties on a right quaternionic Hilbert space. Further,
we shall also demonstrate squeezed states can be defined on the same Hilbert
space, but the noncommutativity of quaternions prevents us in getting the
desired results.Comment: Conference paper. arXiv admin note: text overlap with
arXiv:1704.02946; substantial text overlap with arXiv:1706.0068
de Branges-Rovnyak spaces: basics and theory
For a contractive analytic operator-valued function on the unit disk
, de Branges and Rovnyak associate a Hilbert space of analytic
functions and related extension space
consisting of pairs of analytic functions on the unit disk . This
survey describes three equivalent formulations (the original geometric de
Branges-Rovnyak definition, the Toeplitz operator characterization, and the
characterization as a reproducing kernel Hilbert space) of the de
Branges-Rovnyak space , as well as its role as the underlying
Hilbert space for the modeling of completely non-isometric Hilbert-space
contraction operators. Also examined is the extension of these ideas to handle
the modeling of the more general class of completely nonunitary contraction
operators, where the more general two-component de Branges-Rovnyak model space
and associated overlapping spaces play key roles. Connections
with other function theory problems and applications are also discussed. More
recent applications to a variety of subsequent applications are given in a
companion survey article
Restrictions and extensions of semibounded operators
We study restriction and extension theory for semibounded Hermitian operators
in the Hardy space of analytic functions on the disk D. Starting with the
operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D)
of measure zero, there is a densely defined Hermitian restriction of zd/dz
corresponding to boundary functions vanishing on F. For every such restriction
operator, we classify all its selfadjoint extension, and for each we present a
complete spectral picture.
We prove that different sets F with the same cardinality can lead to quite
different boundary-value problems, inequivalent selfadjoint extension
operators, and quite different spectral configurations. As a tool in our
analysis, we prove that the von Neumann deficiency spaces, for a fixed set F,
have a natural presentation as reproducing kernel Hilbert spaces, with a
Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure
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