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Vijaya Kumar U

Publications and source records attributed to Vijaya Kumar U.

7 recordsLinked to original sources
arXiv2026

On factorization of the shift semigroup

Let $\E$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup $§^\E=(S_t^\E)_{t\ge 0}$ on $L^2(\R_+,\E)$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $(\V_1,\V_2,...,\V_n)$ where $\V_i=(V_{i,t})_{t\ge 0}$ for $i=1,2,...,n$ are semigroups of contractions satisfying $V_{i,t}V_{j,t}=V_{j,t}V_{i,t}$ for all $i$ and $j$ and $S_t^\E=V_{1,t}V_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underline{A}=(A_1,A_2,...,A_n)$ and tuples of positive contractions $\underline{B}=(B_1,B_2,...,B_n)$ on $\E$ satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions \[P:=\{f:\D\to \C \text{ is analytic}, \Re{f}>0 \text{ and }f(0)=1 \}.\]

math.FA↗
arXiv2024

On pure contractive semigroups

We find the commutant of a pure contractive semigroup on a Hilbert space. We demonstrate that any tuple of doubly commuting pure contractive semigroups can be dilated to a tuple of doubly commuting pure isometric semigroups. En route, we obtain a complete model for the tuples of doubly commuting isometric semigroups.

math.FA↗
arXiv2024

Functions with image in a strip

We consider holomorphic functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups.

math.FA↗
arXiv2022

On the structure and the joint spectrum of a pair of commuting isometries

The study of a pair $(V_1,V_2)$ of commuting isometries is a classical theme. We shine new light on it by using the defect operator. In the cases when the defect operator is zero or positive or negative, or the difference of two mutually orthogonal projections with ranges adding up to $\ker (V_1V_2)^*$, we write down structure theorems for $(V_1,V_2)$. The structure theorems allow us to compute the joint spectrum in each of the cases above. Moreover, in each case, we also point out at which stage of the Koszul complex the exactness breaks. A pair of operator valued functions $(φ_1,φ_2)$ is canonically associated by Berger, Coburn and Lebow with $(V_1,V_2)$. If $(V_1,V_2)$ is a pure pair, then in each case above we show that $σ(V_1,V_2)=\bar{\cup_{z\in\D} σ(φ_1(z),φ_2(z))}.$ It has been known that the fundamental pair of commuting isometries with positive defect is the pair of multiplication operators by the coordinate functions on the Hardy space of the bidisc. A major contribution of this note is to figure out the fundamental pair of commuting isometries with negative defect. This pair of commuting isometries is constructed on the Hardy space of the bidisc.

math.FA↗
arXiv2020

Structure of block quantum dynamical semigroups and their product systems

W. Paschke's version of Stinespring's theorem associates a Hilbert $C^*$-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a $C^*$-algebra $\mathcal A$ one may associate an inclusion system $E=(E_t)$ of Hilbert $\mathcal A$-$\mathcal A$-modules with a generating unit $ξ=(ξ_t)$. Suppose $\mathcal B$ is a von Neumann algebra, consider $M_2(\mathcal B)$, the von Neumann algebra of $2\times 2$ matrices with entries from $\mathcal B$. Suppose $(Φ_t)_{t\ge 0}$ with $Φ_t=\begin{pmatrix} ϕ_t^1& ψ_t ψ_t^*&ϕ_t^2 \end{pmatrix},$ is a QDS on $M_2(B)$ which acts block-wise and let $(E^i_t)_{t\ge 0}$ be the inclusion system associated to the diagonal QDS $(ϕ^i_t)_{t\ge 0}$ with the generating unit $(ξ_t^i)_{t\ge 0}, i=1,2.$ It is shown that there is a contractive (bilinear) morphism $T=(T_t)_{t\ge0}$ from $(E^2_t)_{t\ge 0}$ to $(E^1_t)_{t\ge 0}$ such that $ψ_t(a)=\langle ξ^1_t, T_t aξ^2_t\rangle $ for all $a\in\mathcal B.$ We also prove that any contractive morphism between inclusion systems of von Neumann $\mathcal B$-$\mathcal B$-modules can be lifted as a morphism between the product systems generated by them. We observe that the $E_0$-dilation of a block quantum Markov semigroup (QMS) on a unital $C^*$-algebra is again a semigroup of block maps.

math.OA↗
arXiv2019

Roots of Completely Positive Maps

We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous one-parameter semigroups of roots. We present structural and general existence and non-existence results, some special examples in settings where we understand the situation better, and several challenging open problems. Our study is closely related to Elfving's embedding problem in classical probability and the divisibility problem of quantum channels.

math.OA↗