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Shubham Rastogi

Publications and source records attributed to Shubham Rastogi.

6 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↗
arXiv2024

Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$

Douglas and Rudin proved that any unimodular function on the unit circle $\T$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\T$.

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.

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