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K. Thirulogasanthar

Publications and source records attributed to K. Thirulogasanthar.

At least 19 recordsLinked to original sources
arXiv2019

Berberian extension and its S-spectra in a quaternionic Hilbert space

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 extension is that it converts approximate point spectrum of $A$ into point spectrum of $A^\circ$. We show that the same is true for the quaternionic S-spectrum. As in the complex case, we use the Berberian extension to study some properties of the commutator of two quaternionic bounded right linear operators.

math.FA↗
arXiv2019

Perturbation of Continuous Frames on Quaternionic Hilbert Spaces

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.

math.FA↗
arXiv2019

Decomposable operators, local S-spectrum and S-spectrum in the quaternionic setting

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, surjectivity S-spectrum and the union of all local S-spectra coincide. Using continuous right slice-regular functions we have also studied certain properties of local S-spectrum and local S-spectral subspaces.

math.FA↗
arXiv2019

Kato S-spectrum in the quaternionic setting

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 point of a right quaternionic Hilbert space, and the local spectral subsets are introduced and studied. The S-surjectivity spectrum and its connections to the Kato S-spectrum, approximate S-point spectrum and local S-spectrum are investigated. The generalized Kato S-spectrum is introduced and it is shown that the generalized Kato S-spectrum is a compact subset of the S-spectrum.

math.FA↗
arXiv2018

Weyl and Browder S-spectrums in a right quaternionic Hilbert space

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 introduce the Browder S-spectrum.

math-ph↗
arXiv2018

Fredholm operators and essential S-spectrum in the quaternionic setting

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 operator perturbations. Finally, in association with Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the $S$-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel we study left and right S-spectrums as needed for the development of the theory presented in this note.

math.FA↗
arXiv2018

Frame Multipliers for discrete frames on Quaternionic Hilbert Spaces

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 multipliers and weighted frames in the same Hilbert space.

math.FA↗
arXiv2018

Lagrange Multiplier Local Necessary and Global Sufficiency Criteria for Some Non-Convex Programming Problems

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. The quadratic fractional problem is studied with quadratic fractional constraints in mixed variables. For all three problems we reformulate the problem as a mathematical programming problem and apply standard Karush Kuhn Tucker necessary conditions. Then, for each problem, we provide local necessary optimality condition. Further, for each problem a Lagrangian multiplier sufficient optimality condition is provided to identify global minimizer among the local minimizers. For the quadratic problem underestimation of a Lagrangian was employed to obtain the desired sufficient conditions. For the $ρ$-convex problem we obtain two sufficient optimality conditions to distinguish a global minimizer among the local minimizers, one with an underestimation of a Lagrangian and the other with a different technique. A global sufficient optimality condition for the quadratic fractional problem is obtained by reformulating the problem as a quadratic problem and then utilizing the results of the quadratic problem. Examples are provided to illustrate the significance of the results obtained.

math.OC↗
arXiv2018

S-Spectrum and the quaternionic Cayley transform of an operator

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 quaternionic setting.

math-ph↗
arXiv2017

Deficiency Indices of Some Classes of Unbounded $\mathbb{H}$-Operators

In this paper we define the deficiency indices of a closed symmetric right $\mathbb{H}$-linear operator and formulate a general theory of deficiency indices in a right quaternionic Hilbert space. This study provides a necessary and sufficient condition in terms of deficiency indices and in terms of S-spectrum, parallel to their complex counterparts, for a symmetric right $\mathbb{H}$-linear operators to be self-adjoint.

math-ph↗
arXiv2017

On the dimensions of oscillator-like algebras induced by orthogonal polynomials: non-symmetric case

There is a generalized oscillator-like algebra associated with every class of orthogonal polynomials $\{Ψ_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a four term non-symmetric recurrence relation $xΨ_n(x)=b_nΨ_{n+1}(x)+a_nΨ_n(x)+b_{n-1}Ψ_{n-1}(x),~Ψ_0(x)=1,~b_{-1}=0$. This note presents necessary and sufficient conditions on $a_n$ and $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator-like algebras associated with Laguerre and Jacobi polynomials.

math-ph↗
arXiv2017

General construction of Reproducing Kernels on a quaternionic Hilbert space

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 in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

math-ph↗
arXiv2017

A Representation of Weyl-Heisenberg Lie Algebra in the Quaternionic Setting

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 quaternions and on a quaternionic slice, namely on a copy of the complex plane inside the quaternions. For the quaternionic harmonic oscillator, the uncertainty relation is shown to saturate on a neighborhood of the origin in the case we consider the whole set of quaternions, while it is saturated on the whole slice in the case we take the slice-wise approach. In analogy with the complex Weyl-Heisenberg Lie algebra, Lie algebraic structures are developed for the quaternionic case. Finally, we introduce a quaternionic displacement operator which is square integrable, irreducible and unitary, and we study its properties.

math-ph↗
arXiv2017

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.

math-ph↗
arXiv2017

Squeezed states in the quaternionic setting

Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that pure squeezed 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 prevent us in getting the desired results. However, we will show that if once considers the quaternionic slice wise approach, then the desired properties can be obtained for quaternionic squeezed states.

math-ph↗
arXiv2016

Discretization of quaternionic continuous wavelet transforms

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 wavelet transform, living in a complex valued Hilbert space of square integrable functions, of the quaternion wavelet group is discretized, and thereby, a discrete frame for quaternion valued Hilbert space of square integrable functions is obtained.

math-ph↗
arXiv2016

Coherent states on quaternion Slices and a measurable field of Hilbert spaces

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. Hilbert spaces over quaternion slices are identified as representation spaces for a set of irreducible unitary group representations and their direct integral is shown to be a reducible representation for the Hilbert space over the whole quaternion field.

math-ph↗