Thirulogasanthar Kengatharam

Concordia University Montreal · Department of Computer Science and Software Engineering

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local S-spectrum, at a...

For a bounded right linear operators A, in a right quaternionic Hilbert space \(V_\mathbb {H}^R\), following the complex formalism, we study the Berberian extension \(A^\circ \), which is an extension of A in a right quaternionic Hilbert space obtained from \(V_\mathbb {H}^R\). In the complex setting, the important feature of the Berberian extensio...

Using a left multiplication defned on a right quaternionic Hilbert space, we shall demonstrate that pure squeezed states can be defned with all the desired properties on a right quaternionic Hilbert space. Further, we shall also demonstrate squeezed states can be defned on the same Hilbert space, but the noncommutativity of quaternions prevent us i...

For a bounded right linear operators A, in a right quaternionic Hilbert space V, following the complex formalism, we study the Berberian extension A' , which is an extension of A in a right quaternionic Hilbert space obtained from V. In the complex setting, the important feature of the Berberian extension is that it converts approximate point spect...

In this note, following the theory of discrete frame perturbations in a complex Hilbert space, we examine perturbation of rank $n$ continuous frame, rank $n$ continuous Bessel family and rank $n$ continuous Riesz family in a non-commutative setting, namely in a right quaternionic Hilbert space.

In a right quaternionic Hilbert space, following the complex formalism, decomposable operators, the so-called Bishop's property and the single valued extension property are defined and the connections between them are studied to certain extent. In particular, for a decomposable operator, it is shown that the S-spectrum, approximate S-point spectrum...

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local S-spectrum, at a...

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate S-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact oper...

In this note first we study the Weyl operators and Weyl S-spectrum of a bounded right quaternionic linear operator, in the setting of the so-called S-spectrum, in a right quaternionic Hilbert space. In particular, we give a characterization for the S-spectrum in terms of the Weyl operators. In the same space we also study the Browder operators and...

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 which are obtained by the sole action of the squeeze operator on the vacuum state, fermionic coherent states can be defined with all the desired propertie...

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate S-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact oper...

In this note, following the complex theory, we examine discrete controlled frames, discrete weighted frames and frame multipliers in a non-commutative setting, namely in a left quaternionic Hilbert space. In particular, we show that the controlled frames are equivalent to usual frames under certain conditions. We also study connection between frame...

In this paper we consider three minimization problems, namely quadratic, ρ-convex and quadratic fractional programing problems. The quadratic problem is considered with quadratic inequality constraints with bounded continuous and discrete mixed variables. The ρ-convex problem is considered with ρ-convex inequality constraints in mixed variables. Th...

In this paper we consider three minimization problems, namely quadratic, $\rho$-convex and quadratic fractional programing problems. The quadratic problem is considered with quadratic inequality constraints with bounded continuous and discrete mixed variables. The $\rho$-convex problem is considered with $\rho$-convex inequality constraints in mixe...

In this paper we define the quaternionic Cayley transformation of a densely defined, symmetric, quaternionic right linear operator and formulate a general theory of defect number in a right quaternionic Hilbert space. This study investigates the relation between the defect number and S-spectrum, and the properties of the Cayley transform in the qua...

Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with their complex counterpart. With the aid of the so-obtained position and momentum operators, we study the Heisenberg uncertainty principle on the whole set of qua...

In this paper we define the deficiency indices of a closed symmetric right H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb H$$\end{document}-linear operator an...

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 squeez...

A scheme to form a basis and a frame for a Hilbert space of quaternion valued square integrable function from a basis and a frame, respectively, of a Hilbert space of complex valued square integrable functions is introduced. Using the discretization techniques for 2D-continuous wavelet transform of the SIM (2) group, the quaternionic continuous wav...

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 squeez...

A set of reproducing kernel Hilbert spaces are obtained on Hilbert spaces over quaternion slices with the aid of coherent states. It is proved that the so obtained set forms a measurable field of Hilbert spaces and their direct integral appears again as a reproducing kernel Hilbert space for a bigger Hilbert space over the whole quaternions. Hilber...

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator valued measures is proved...

There is a generalized oscillator-like algebra associated with every class of orthogonal polynomials $\{\Psi_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a four term non-symmetric recurrence relation $x\Psi_n(x)=b_n\Psi_{n+1}(x)+a_n\Psi_n(x)+b_{n-1}\Psi_{n-1}(x),~\Psi_0(x)=1,~b_{-1}=0$. This note presents necessary and sufficient conditions...

A class of orthogonal polynomials in two quaternionic variables is introduced. This class serve as an analogous to the classical Zernike polynomials (arXiv: 1502.07256, 2014). A number of interesting properties such as the orthogonality condition, recurrence relations, raising and lowering operators are discussed in details. Particularly, the ladde...

As needed for the construction of rank $n$ continuous frames on a right quaternionic Hilbert space the so-called S-spectrum of a right quaternionic operator is studied. Using the S-spectrum, as for the case of complex Hilbert spaces, along the lines of the arguments of {\em Ann.Phys.}, {\bf 222} (1993), 1-37., various classes of rank $n$ continuous...

Parallel to the quan-tization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quater-nionic quantum mechanics is quantized and using the quanti-zation the position and momentum operators are obtained by us in [1]. In this article, we show that the right quaternionic canonical cohe...

A general theory of frames on finite dimensional quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. In this paper we established the frame decomposition theorem for quaternion Hilbert space. At the conclusion a perceptive clarification of why frames are important in signal transmission is given.

Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols and related quantities are analysed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are al...

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilb...

There is a generalized oscillator algebra associated with every class of orthogonal polynomials $\{\Psi_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a three term recurrence relation $x\Psi_n(x)=b_n\Psi_{n+1}(x)+b_{n-1}\Psi_{n-1}(x), \Psi_0(x)=1, b_{-1}=0$. This note presents necessary and safficient conditions on $b_n$ for such algebras to...

Using the orthonormality of the 2D-Zernike polynomials reproducing kernels, reproducing kernel Hilbert spaces and ensuing coherent states are attained. With the aid of the so obtained coherent states the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms are also obtained. Along the wa...

A general theory of frames on finite dimensional quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.

Temporally stable coherent states are discussed for an abstract Hamiltonian with a general spectrum. Statistical quantities related to the coherent states are calculated. As special cases of the construction, coherent states for some well-known Hamiltonians, namely; Harmonic oscillator, Isotonic oscillator, pseudoharmonic oscillator, Infinite well...

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic $L^2$-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these $L^2$-spaces, the associated reproducin...

Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117–26) it is proved that these coher...

We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analog...

The quaternionic vector coherent states are realized as coherent states of the supersymmetric harmonic oscillator with broken symmetry in analogy with the standard canonical coherent states of the ordinary harmonic oscillator. We study the nonclassical properties of the oscillator, such as the photon number distribution and signal-to-quantum-noise...

The well-known canonical coherent states are expressed as infinite series in powers of a complex number z and a positive integer ρ(m) = m!. In analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable z with a real Clifford matrix. We also present another class of vector coherent stat...

We present a general scheme for tensoring matrices and thereby, for Clifford type matrices, generate new classes of vector coherent states. Properties of these vector coherent states are analyzed in similarity with the well-known canonical coherent states. As examples, tensored quaternions and octonions are discussed.

Eigenfunctions and eigenvalues of the free magnetic Schrödinger operator, describing a spinless particle confined to an infinite layer of fixed width, are discussed in detail. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space. Four different classes of temporally stable coherent states associated with the operator...

A frame is an overcomplete family of vectors in a Hilbert space in which the orthogonality condition is relaxed. The Julia set is the chaotic regime of a rational function. In this note, we label frames of an abstract Hilbert space by elements of the Julia set of a rational function.

Frames of finite dimensional Hilbert spaces have recently been of great interest in applications to modern communication networks transport packets. In this note, continuous and discrete frames, living on fractal sets, of both finite and infinite dimensional separable abstract Hilbert spaces are found. In particular, we find discrete frames, robust...

Classes of coherent states are presented by replacing the labeling parameter $z$ of Klauder-Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mitta...

Canonical coherent states can be written as infinite series in powers of a single complex number $z$ and a positive integer $\rho(m)$. The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers $\{\rho(m)\}_{m=0}^\infty$. In this paper we...

In this paper, we discretize the continuous theory of coherent states on a general semidirect product group G = V S, where V is a vector space and S ⊂ GL(V) is a semisimple connected Lie group. We show that it is always possible to construct a discrete frame associated with a unitary irreducible representation of G, which is square integrable modul...

The well-known canonical coherent states are expressed as an infinite series in powers of a complex number $z$ together with a positive sequence of real numbers $\rho(m)=m$. In this article, in analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable $z$ by a real Clifford matrix. We...

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. F...

The canonical coherent states are expressed as infinite series in powers of a complex number $z$ in their infinite series version. In this article we present classes of coherent states by replacing this complex number $z$ by other choices, namely, iterates of a complex function, higher functions and elementary functions. Further, we show that some...

We present a class of vector coherent states in the domain $D\times D\times >....\times D$ (n-copies), where $D$ is the complex unit disc, using a specific class of hermitian matrices. Further, as an example, we build vector coherent states in the unit disc by considering the unit disc as the homogeneous space of the group SU(1,1).

A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable $z$. In the present scheme, the variable $z$ is replaced by a matrix valued function over appropriate domains....

We develop a method of discretization of the continuous theory of coherent states on a general semidirect product Lie group. The group is assumed to have a unitary representation which is square integrable on some homogeneous space. We show also that the existence of a discrete frame of coherent states in the carrier space of a unitary representati...

In this thesis we develop vector coherent states (VCS) in the form [Special characters omitted.] where [Special characters omitted.] , the n dimensional complex space and [Special characters omitted.] is an n x n matrix. By imposing some conditions on the matrix [Special characters omitted.] we develop a general procedure to obtain VCS. We develop...