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P. Santhosh Kumar

Publications and source records attributed to P. Santhosh Kumar.

7 recordsLinked to original sources
arXiv2021

Stinespring's Theorem for Unbounded Operator valued Local completely positive maps and Its Applications

Anar A. Dosiev in [Local operator spaces, unbounded operators and multinormed $C^*$-algebras, J. Funct. Anal. 255 (2008), 1724-1760], obtained a Stinespring's theorem for local completely positive maps (in short: local CP-maps) on locally $C^{\ast}$-algebras. In this article a suitable notion of minimality for this construction has been identified so as to ensure uniqueness up to unitary equivalence for the associated representation. Using this a Radon-Nikodym type theorem for local completely positive maps has been proved. Further, a Stinespring's theorem for unbounded operator valued local completely positive maps on Hilbert modules over locally $C^{\ast}$-algebras (also called as local CP-inducing maps) has been presented. Following a construction of M. Joiţa, a Radon-Nikodym type theorem for local CP-inducing maps has been shown. In both cases the Radon-Nikodym derivative obtained is a positive contraction on some complex Hilbert space with an upward filtered family of reducing subspaces.

math.OA↗
arXiv2019

Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $δ>0$ there exist a compact operator $K$ with $\|K\|< δ$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not be normal) is disconnected by a pair of disjoint axially symmetric closed subsets, then it is strongly reducible.

math.FA↗
arXiv2018

A note on convexity of sections of quaternionic numerical range

The quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. We prove Toeplitz-Hausdorff like theorem, that is, for any given quaternionic matrix every section of its quaternionic numerical range is convex. We provide some additional equivalent conditions for the quaternionic numerical range of matrices over quaternions to be convex and prove some numerical radius inequalities.

math.FA↗
arXiv2017

Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces

In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let $T$ be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert space $H$ with domain $\mathcal{D}(T)$, a right linear subspace of $H$ and fix a unit imaginary quaternion, say $m$. Then there exists a Hilbert basis $\mathcal{N}$ of $H$ and a unique quaternionic spectral measure $F$ on the $σ$- algebra of $\mathbb C_m^{+}$ (upper half plane of the slice complex plane $\mathbb C_m$) associated to $T$ such that \begin{equation*} \left\langle x | Ty \right\rangle = \int\limits_{σ_{S}(T) \cap \mathbb{C}_{m}^{+}}λ\ dF_{x,y}(λ),\; \text{ for all}\; y \in \mathcal{D}(T),\ x \in H, \end{equation*} where $F_{x,y}$ is a quaternion valued measure on the $σ$- algebra of $\mathbb{C}_{m}^{+}$, for any $x,y\in H$ and $σ_{S}(T)$ is the spherical spectrum of $T$. Here the representation of $T$ is established with respect to the Hilbert basis $\mathcal{N}$. To prove this result, we reduce the problem to the complex case and obtain the result by using the classical result.

math.SP↗
arXiv2017

Spectral Theorem for quaternionic normal operators: Multiplication form

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis $\mathcal{N}_{m}$ of $\mathcal{H}$, a measure space $(Ω, μ)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(Ω; \mathbb{H}; μ)$ and a $μ$ - measurable function $ϕ\colon Ω\to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{α+ m β; \;α, β\in \mathbb{R}\}$) such that \[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_ϕ$ is the multiplication operator on $L^{2}(Ω; \mathbb{H}; μ)$ induced by $ϕ$ with $ U(\mathcal{D}(T)) \subseteq \mathcal{D}(M_ϕ)$. In the process, we prove that every complex Hilbert space is a slice Hilbert space. We establish these results by reducing it to the complex case then lift it to the quaternionic case.

math.SP↗
arXiv2015

On the polar decomposition of right linear operators in quaternionic Hilbert spaces

In this article we prove the existence of the polar decomposition for densely defined closed right linear operators in quaternionic Hilbert spaces: If $T$ is a densely defined closed right linear operator in a quaternionic Hilbert space $H$, then there exists a partial isometry $U_{0}$ such that $T = U_{0}|T|$. In fact $U_{0}$ is unique if $N(U_{0}) = N(T)$. In particular, if $H$ is separable and $U$ is a partial isometry with $T = U|T|$, then we prove that $U = U_{0}$ if and only if either $N(T) = \{0\}$ or $R(T)^{\bot} = \{0\}$.

math.FA↗