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

Positive and generalized positive real lemma for slice hyperholomorphic functions

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball

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

Boundary interpolation for slice hyperholomorphic Schur functions

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad {\rm for} \quad u=1,\ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces

Phase Coexistence Near a Morphotropic Phase Boundary in Sm-doped BiFeO3 Films

We have investigated heteroepitaxial films of Sm-doped BiFeO3 with a Sm-concentration near a morphotropic phase boundary. Our high-resolution synchrotron X-ray diffraction, carried out in a temperature range of 25C to 700C, reveals substantial phase coexistence as one changes temperature to crossover from a low-temperature PbZrO3-like phase to a high-temperature orthorhombic phase. We also examine changes due to strain for films greater or less than the critical thickness for misfit dislocation formation. Particularly, we note that thicker films exhibit a substantial volume collapse associated with the structural transition that is suppressed in strained thin films

de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. 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 ${\mathcal H}(S)$, 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 ${\mathcal D}(S)$ 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