Inner multipliers and Rudin type invariant subspaces

Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ be the $\mathcal{E}$-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well known Beurling-Lax-Halmos theorem states that every shift invariant subspace of $H^2_{\mathcal{E}}(\mathbb{D})$ other than $\{0\}$ has the form $\Theta H^2_{\mathcal{E}_*}(\mathbb{D})$, where $\Theta$ is an operator-valued inner multiplier in $H^\infty_{B(\mathcal{E}_*, \mathcal{E})}(\mathbb{D})$ for some Hilbert space $\mathcal{E}_*$. In this paper we identify $H^2(\mathbb{D}^n)$ with $H^2(\mathbb{D}^{n-1})$-valued Hardy space $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ and classify all such inner multiplier $\Theta \in H^\infty_{\mathcal{B}(H^2(\mathbb{D}^{n-1}))}(\mathbb{D})$ for which $\Theta H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ is a Rudin type invariant subspace of $H^2(\mathbb{D}^n)$.Comment: 8 page

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case

The Defect Sequence for Contractive Tuples

We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.Comment: 16 Pages. To appear in Linear Algebra and its Application

On certain Toeplitz operators and associated completely positive maps

We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to $L^\infty$ of a certain compact subset of $\mathbb C^n$. Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do not hold in general for $n>2$.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1706.0346

NDVI and NDWI based Change Detection Analysis of Bordoibam Beelmukh Wetlandscape, Assam using IRS LISS III data

This paper analyses per pixel change detection of Bordoibam Beelmukh wetlandscape located in Dhemaji district of Assam, India, which covers around 23 sq. km of geographical area. Normalized Difference Vegetation Index (NDVI) and Normalized Difference Water Index (NDWI) are the two remote sensing indices applied for change detection analysis in the study area. IRS LISS III satellite imageries with 23.5 meter spatial resolution have been used to conduct the analysis. These satellite imageries have been collected from NRSC Bhuvan portal with 5 years temporal interval (2008 to 2013). Image differencing technique has been applied to detect per pixel change using NDVI and NDWI difference image results of the wetlandscape. The study has observed per pixel change detection in five distinct categories for both NDVI and NDWI results of the study area. These are increased more than 5 percent, decreased more than 5 percent, some increase, some decrease and unchanged. In this regard, the study reveals that there is maximum change (79% to total change) in increased more than 5 percent category for NDVI, whereas for NDWI maximum change (96% to total change) is observed under decreased more than 5 percent category of the study area. It has been also observed that there are significant changes of both NDVI and NDWI values from 2008 to 2013 in the study area which in turn indicate changes of vegetation and water cover areas of the same