Project
financed by UEFISCDI - Romanian Government

Code: PN-II-ID-PCE-2011-3-0119

Contract 73/2011

Multidimensional operator theory: reproducing kernel spaces, noncommutative functions, moment problems, and generalized dilation results

Code: PN-II-ID-PCE-2011-3-0119

Contract 73/2011

Multidimensional operator theory: reproducing kernel spaces, noncommutative functions, moment problems, and generalized dilation results

Summary
of the project

The
theory
of a single operator on Hilbert space has been thoroughly developed in the
last decades. Recent perspectives stem from the emergent area of
multioperators; one can consider it as an analogue of studying functions of
several variables instead of a single variable. The project attempts to
explore several directions in this domain. These include: reproducing kernel
spaces that have the Pick property, noncommutative functions and their
applications, dilation theorems in vector Hilbert spaces, multidimensional
operator valued moment problems, multivariate prediction theory.Research Team:

- Dr. Dan Grigore Timotin (project director)
- Dr. Aurelian Gheondea
- Dr. Ilie Valusescu
- Dr. Calin Grigore Ambrozie
- Dr. Mihai Popa
- Dr. Bebe Prunaru

- Study of reproducing kernel spaces that have the Pick property.
- Study of noncommutative functions.
- Dilation theorems in vector Hilbert spaces.
- Moment problems in several variables.
- Dilation theory and operator valued processes.

Final scientific report: 2011-2016

Scientific report - 2015

Scientific report - 2014

Scientific report - 2011-2013

Papers

Published:

- Multivariate
truncated moments problems and maximum entropy

- Factorizations
of analytic self-maps of the upper half-plane

- A
note on composition operators in a half-plane

- Dilations
of some VH-spaces operator valued invariant kernels

- Notes on continuous parameter periodically Gamma-correlated processes
- The numerical range of a contraction with finite defect numbers
- A Poisson boundary for topological semigroups
- Asymptotic stability for a class of Markov semigroups
- Schur coupling and related equivalence relations for operators on a Hilbert space
- Some geometrical aspects of the Gamma-correlated processes
- Nonextreme de Branges-Rovnyak spaces as models for contractions
- Real
second order freeness and Haar orthogonal matrices

- Some remarks on the infinite variate prediction (in Proceedings Volume)
- Commutation relations for truncated Toeplitz operators
- On fluctuation of traces of large matrices over a non-commutative algebra
- Contractively included subspaces of Pick spaces
- An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization
- Triplets of closely embedded Hilbert spaces
- Representations of *-semigroups associated to invariant kernels with values adjointable operators
- A survey of reproducing kernel Krein spaces (book chapter)
- Very short introduction to de Branges-Rovnyak spaces (book chapter)
- Some remarks on the infinite variate prediction II (in Proceedings Volume)
- Continuous parameter Gamma-stationary processes (in Proceedings Volume)
- Stationary
dilation of a nonstationary Gamma-correlated process (in
Proceedings Volume)

Submitted:

- Operators models for
Hilbert locally C*-modules

- Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators

Preprints:

- A
short proof of a Choquet-Deny theorem for abelian topological
semigroups

- On a model for the maximal function of an n-hypercontraction
- On the relation between complex and real second order free independence
- Freeness and the transposes of unitarily invariant random matrices
- On the multiplication of operator-valued c-free random variables
- Interpolation for completely positive maps: numerical solutions
- H
^{2}-spaces of non-commutative functions