Ajay Kumar

Ajay Kumar
University of Delhi | DU · Department of Mathematics (Faculty of Mathematical Sciences )

PhD

About

111
Publications
7,168
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524
Citations
Citations since 2016
33 Research Items
183 Citations
2016201720182019202020212022010203040
2016201720182019202020212022010203040
2016201720182019202020212022010203040
2016201720182019202020212022010203040
Additional affiliations
December 2012 - December 2015
University of Delhi
Position
  • Head of Department

Publications

Publications (111)
Article
We prove a necessary and sufficient condition for embeddability of an operator system into $\mO_2$. Using Kirchberg's theorems on tensor product of $\mO_2$ and $\mO_{\infty}$, we establish results on their operator system counterparts $\mS_2$ and $\mS_{\infty}$. Applications of the results proved, including some examples describing $C^*$-envelopes...
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We study the relationship between $C^*$-envelopes and inductive limit of operator systems. Various operator system nuclearity properties of inductive limit for a sequence of operator systems are also discussed.
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We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner, that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $\lambda$ for the algebraic tensor...
Article
For higher order complex partial differential equations a variety of boundary value problems are investigated. They are compositions of the three basic boundary value problems in complex analysis, the Schwarz, the Dirichlet and the Neumann problem. Only some of them are well-posed. For the others being overdetermined solvability conditions are atta...
Chapter
We define possible and recurrent elements for random walks on commutative hypergroups and show that for a large class of random walks, either no elements is recurrent or all possible elements are recurrent and they form a closed subhypergroup.
Article
We initiate a study of ternary rings of unbounded operators (TRUOs) which are local version of ternary rings of bounded operators. Abstract definition of ternary Pro \(C^*\)-rings is also proposed. A one-to-one correspondence between representations of TRUO and its linking Pro \(C^*\)-algebra is obtained. Finally we examine their tensor products pa...
Article
Local operator systems are projective limits of operator systems. In this paper, we discuss several tensor products including minimal (lmin), maximal (lmax), local commuting, and λ-tensor product in the category of local operator systems. A characterization of (lmin, lmax)-nuclearity is given which is local version of approximation theorem. We also...
Article
We define quantized Hilbert modules over local operator algebras which coincide with continuous representations of local operator algebras on a quantized domain. It is shown that for a local operator algebra A, the restriction \(\rho \restriction _A\) of a representation \(\rho\) of \(C^*(A)\) on a quantized domain of closed subspaces of Hilbert sp...
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Given two operator spaces E and F, injectivity of the canonical map from the \(\lambda \)-tensor product \(E\bigotimes ^\lambda F\) into the operator space injective tensor product \(E\bigotimes ^{\min } F\) is investigated, where the \(\lambda \)-tensor product is a generalization of the Haagerup, the operator space projective and the Schur tensor...
Article
The present article endeavours to develop partial optional randomized - response technique (PORT) to deal with sensitive issues in presence of non-response in successive sampling. Calibration techniques have been embedded with PORT to estimate sensitive population mean at current move in two move successive sampling in presence of non-response. Opt...
Article
We introduce and explore the theory of tensor products in the category of local operator systems. Analogous to minimal operator system OMIN and maximal operator system OMAX, minimal and maximal local operator system structures LOMIN and LOMAX, respectively, are also discussed.
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Estimates are obtained for the initial coefficients of a normalized analytic function f in the unit disk D such that f and the analytic extension of f-1 to D belong to certain subclasses of univalent functions. The bounds obtained improve some existing known bounds.
Preprint
We introduce and explore the theory of tensor products in the category of local operator systems. Analogous to minimal operator system OMIN and maximal operator system OMAX, minimal and maximal local operator system structures LOMIN and LOMAX, respectively, are also discussed.
Article
The support of a wavelet transform associated with a square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Assumptions are illustrated and supported by examples. The pointwise homogeneous approximation property for a wavelet transform has been investigated. An analogue of Heisenberg type inequality i...
Preprint
Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent functions. The bounds obtained improve some existing known bounds.
Article
A normalized analytic function f is lemniscate starlike if the quantity \(zf'(z)/f(z)\) lies in the region bounded by the right half of the lemniscate of Bernoulli \(|w^2-1|=1\). It is Janowski starlike if the quantity \(zf'(z)/f(z)\) lies in the disk whose diametric end points are \((1-A)/(1-B)\) and \((1+A)/(1+B)\) for \(-1\le B<A\le 1\). The rad...
Article
Estimation of sensitive issues is an adversely challenging task as it is highly influenced by social desirability bias leading to untrue response many times. Hence, the possible way out for these problems are randomized response technique or scrambled response technique or the item sum technique etc. The proposed work is a methodological advancemen...
Article
Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and “normalized” Bessel function u p ( z ) = u p,b,c ( z ) satisfies the inequalities ∣(1 + ( zu ″ p ( z )/ u ′ p ( z ))) ² − 1∣ < 1 or ∣(( zup ( z ))′/ u p ( z )) ² − 1∣ < 1. We also determine the condition on these parameters so that . Relations between...
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We study inductive limit in the category of Ternary Ring of oper-ator(TRO). The existence of inductive limit in this category is proved and its behaviour with quotienting is discussed. We investigate whether the linking C *-algebra commutes with inductive limits, in the sense that if (V n , f n) is an inductive system then whether lim → A(V n) = A(...
Article
The problem of estimation of sensitive population mean has been investigated using the item sum technique (IST) with calibration estimators in two occasion successive sampling. Generic sampling design have been assumed at each occasion to define calibration estimators. Properties of the proposed estimators has been discussed including asymptotic va...
Preprint
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The radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1$, have been determined for normalizations of $q$-Bessel function, Bessel function of first kind of order $\nu$, Lommel function of first kind and Legendre polynomial of odd degree.
Preprint
Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(zu''_p(z)/u'_p(z)))^2-1|<1$ or $|((zu_p(z))'/u_p(z))^2-1|<1$. We also determine the condition on these parameters so that $-(4(p+(b+1)/2)/c)u'_p...
Preprint
The support of wavelet transform associated with square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Pointwise homogeneous approximation property for wavelet transform has been investigated. An analogue of Heisenberg type inequality has been also obtained for wavelet transform
Preprint
Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups. Finally Beurling's theorem for Gabor transform is discussed for groups of the form $\mathbb{R}_n \times K$, where $...
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For an analytic function f on the unit disk D = {z : |z
Article
We investigate whether the λ-tensor product of operator spaces which includes the operator space projective, Haagerup and Schur tensor products, and the λ-tensor product of operator systems commute with inductive limits, in the sense that if (Xn,ϕn) is an inductive system of operator spaces (or operator systems) with an inductive limit X, and if Y...
Preprint
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For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to...
Article
We introduce weighted cb maps and $\Lambda_\mu$-cb maps on operator spaces which are generalizations of completely bounded maps and a certain class of bilinear maps on operator spaces which we call $\lambda_\mu$-cb bilinear maps. Some basic properties of these maps and an operator space tensor product associated to $\lambda_\mu$-cb bilinear maps ha...
Article
Classes of locally compact groups having qualitative uncertainty principle for Gabor transform have been investigated. These include Moore groups, Heisenberg Group $\mathbb{H}_n, \mathbb{H}_{n} \times D,$ where $D$ is discrete group and other low dimensional nilpotent Lie groups.
Article
We study the polyharmonic Neumann and mixed boundary value problems on the Korányi ball in the Heisenberg group . Necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem and Neumann–Dirichlet boundary value problems.
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Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form $\mathbb{R}^{n}\times K$ , where $K$ is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group $G...
Article
The existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian relative to the Korányi ball on the Heisenberg group \(\mathbb {H}_{n}\) are discussed. Explicit representation for a Green’s type function (Neumann function) for the Korányi ball in \(\mathbb {H}_{n}\) for circular functions has been obtained. This function...
Article
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We give an existence proof for a polynomial solution of the Poisson equation $L_0 u=q$ where $q$ is a polynomial in the one dimensional Heisenberg Group. All the polynomial solutions of the polyharmonic equation $L_0^m u=0$ in terms of harmonic polynomials are determined. In addition, we also discuss the polyharmonic Neumann and mixed boundary valu...
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We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Korányi ball and for the slice of the Korányi ball in the Heisenberg group H n by obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry.
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We discuss the Qualitative Uncertainty Principle for Gabor transform on certain classes of the locally compact groups, like abelian groups, $\mathbb{R}^n\times K$, $K \ltimes \mathbb{R}^n$ where $K$ is compact group. We shall also prove a weaker version of Qualitative Uncertainty Principle for Gabor transform in case of compact groups.
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We discuss Heisenberg uncertainty inequality for groups of the form $K \ltimes \mathbb{R}^n$, $K$ is a separable unimodular locally compact group of type I. This inequality is also proved for Gabor transform for several classes of groups of the form $K \ltimes \mathbb{R}^n$.
Article
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We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for R n × K (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which includ...
Article
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We propose a theory of $\lambda$-tensor product of operator spaces which extends the theory of Blecher-Paulsen and Effros-Ruan for the operator space projective tensor product \cite{blecp}, \cite{effros}, \cite{eff} and that of Rajpal-Kumar-Itoh for the Schur tensor product \cite{vandee4} of operator spaces.
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We propose a theory of λ-tensor product of operator spaces which extends the theory of Blecher-Paulsen and Effros-Ruan for the operator space projective tensor product [3], [7], [8] and that of Rajpal-Kumar-Itoh for the Schur tensor product [21] of operator spaces.
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Existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian on the Kor\'anyi ball of the Heisenberg group $\mathbb{H}_n$ are discussed. Explicit representations of Green's type function (Neumann function) for the half space and Kor\'anyi ball in $\mathbb{H}_n$ for circular functions have been obtained. These functions are...
Article
We investigate locally compact topological groups for which a generalized analogue of Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\R^n \times K$ (where $K$ is a compact group), Euclidean Motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie gro...
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We develop a systematic study of the schur tensor product both in the category of operator spaces and in that of $C^*$-algebras.
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For $C^*$-algebras $A$ and $B$, we study the bi-continuity of the canonical embedding of $A^{**}\ot_{\gamma} B^{**}$ ($A^{**}\hat{\ot} B^{**}$) into $(A \ot_{\gamma} B)^{**}$ (resp. $(A \hat{\ot} B)^{**}$), and its isomorphism. Ideal structure of $A\hat{\ot} B$ has been obtained in case $A$ or $B$ has only finitely many closed ideals.
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The Banach $^{*}$-algebra $A\hat{\otimes}B$, the operator space projective tensor product of $C^{*}$-algebras $A$ and $B$, is shown to be $^{*}$-regular if Tomiyama's property ($F$) holds for $A\otimes_{\min}B$ and $A \otimes_{\min}B=A \otimes_{\max}B$, where $\otimes_{\min}$ and $\otimes_{\max}$ are the injective and projective $C^{*}$-cross norm,...
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For completely contractive Banach algebras $A$ and $B$ (resp. operator algebras $A$ and $B$), necessary and sufficient conditions for the operator space projective tensor product $A\widehat{\otimes}B$ (resp. the Haagerup tensor product $A\otimes^{h}B$) to be Arens regular are obtained. Using the non-commutative Grothendieck's inequality, we show th...
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We obtain explicit smooth Green's functions for annular domain and infinite strip by using kelvin $R$-transform in the Heisenberg group $\H_n$.
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We give an explicit, and geometrical formula for the fundamental solution for higher order sub-Laplacians on a model step two nilpotent Lie group.
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We study the spectral synthesis for the Banach ∗-algebra A \widehat \oop B, the operator space projective tensor product of C∗ -algebras A and B. It is shown that if A or B has finitely many closed ideals, then A⊗B obeys spectral synthesis. The Banach algebra A⊗A with the reverse involution is also studied.
Article
In this article, a modified Kelvin transform on n using inversion with respect to a ball of arbitrary radius is defined, which gives explicit expressions for Green's function and Poisson's kernel for the Korányi ball of arbitrary radius and annular domain. The solution of the Dirichlet problem for the union of two balls is discussed using the Schwa...
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For $C^{*}$-algebras $A$ and $B$, the operator space projective tensor product $A\hat{\otimes}B$ and the Banach space projective tensor product $A\otimes_{\gamma}B$ are shown to be symmetric. We also show that $A\hat{\otimes}B$ is weakly Wiener algebra. Finally, quasi-centrality, and the unitary group of $A\hat{\otimes}B$ are discussed.
Article
For $C^*$-algebras $A$ and $B$, we prove the slice map conjecture for ideals in the operator space projective tensor product $A \hat\otimes B$. As an application, a characterization of prime ideals in the Banach $\ast$-algebra $A\hat\otimes B$ is obtained. Further, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\h...
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For $C^*$-algebras $A$ and $B$, we prove the slice map conjecture for ideals in the operator space projective tensor product $A \hat\otimes B$. As an application, a characterization of prime ideals in the Banach $\ast$-algebra $A\hat\otimes B$ is obtained. Further, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\h...
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We prove that for operator spaces $V$ and $W$, the operator space $V^{**}\otimes_h W^{**}$ can be completely isometrically embedded into $(V\otimes_h W)^{**}$, $\otimes_h$ being the Haagerup tensor product. It is also shown that, for exact operator spaces $V$ and $W$, a jointly completely bounded bilinear form on $V\times W$ can be extended uniquel...
Article
The equation for , is investigated in the upper half plane. A Cauchy integral representation formula is obtained explicitly. Different forms of boundary conditions originating from the well-known Schwarz, Dirichlet and Neumann problems from complex analysis are studied. These boundary value problems are solved in the upper half plane.
Article
Explicit expressions for the Green´s function and Poisson kernel for quarter space and octants in the Heisenberg group H1 are obtained for circular boundary data. Using Poisson kernel, inhomogeneous Dirichlet problem is discussed on H1. Explicit Green´s function for the powers of Laplacian in case of Korányi ball is also given.
Article
Explicit representations for the solution of higher order equations are obtained for Dirichlet boundary conditions and Schwarz boundary conditions in the upper half plane. We also investigate a mixed boundary value problem for the inhomogeneous polyanalytic equation.
Article
For C*-algebras A and B, the identity map from A [^(Ä)] BA \widehat{\otimes} B into A Ä\otimes λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A⊗B of two C*-algebras A and B with centers Z(A) and Z(B) under operator space projective norm is equal to Z(A)[^(Ä)]Z(B)Z(A)\widehat{\otimes}Z(B) . A ch...
Article
The Green's function and the Poisson kernel are known for the sub-Laplacian on the Koranyi ball in the Heisenberg group for circular boundary data. We present a representation formula for higher powers of the sub-Laplacian by defining Green's function and Poisson kernel of higher order for certain H-type groups which include the Heisenberg group.
Article
Based on the explicit form of the Green function for Δ n in the unit disc of the complex plane and certain recursive relations, we solve the Dirichlet Problem for the n-Poisson equation.
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The Neumann boundary value problem is investigated for the inhomogeneous polyanalytic equation. Then mixed k-Neumann and n-k Schwarz boundary value problem is also studied.
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Explicit representations for the solutions of the mixed boundary value problems arising from p-Schwarz, q-Dirichlet and r-Neumann for the inhomogeneous polyanalytic equation have been obtained.
Article
Further mixed boundary value problems are studied for the inhomogeneous polyanalytic equation in the unit disc. As in part I, see [4], the boundary conditions are some combinations of Schwarz, Dirichlet and Neumann conditions. The method applied is based on an iteration process leading from lower order equations to higher ones, see [1-3]. The metho...
Article
The theory of bi-analytic functions introduced by Hua, Lin and Wu in the 1980s in order to solve some second-order systems of two partial differential equations in two variables is the theory of the second-order complex partial differential equation for some constant real α. Here the equation is investigated for 1 ≤ m, n. In the case m = 1 the solu...
Article
The mixed boundary value problems arising from m-Schwarz, n-Neumann; n-Neumann, m-Dirichlet; m-Dirichlet, n-Schwarz boundary conditions for the inhomogeneous polyanalytic equation have been studied.
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Explicit representations of the solution to higher-order Poisson equation with Dirichlet boundary conditions are obtained. The equations for 1 ≤ m, n have also been investigated.
Article
The three basic boundary value problems in complex analysis are of Schwarz, of Dirichlet and of Neumann type. When higher order equations are investigated all kind of combinations of these boundary conditions are proper to determine solutions. However, not all of these conditions are leading to well-posed problems. Some are overdetermined so that s...
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We extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on n and Heisenberg groups.
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We prove an analogue of Hardy's Theorem for Fourier transform pairs in [open face R] for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on [open face R]n and the Heisenberg groups [open face H]n.
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For a finite dimensional -algebra A and any -algebra B, we determine a constant of equivalence of operator space projective norm and the Banach space projective norm on . We also discuss the *-Banach algebra .
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We show that the involution $\theta(a\otimes b)=a^*\otimes b^*$ on the Haagerup tensor product $A\otimes_{\mrm{H}}B$ of $C^*$-algebras $A$ and $B$ is an isometry if and only if $A$ and $B$ are commutative. The involutive Banach algebra $A\otimes_{\mrm{H}}A$ arising from the involution $a\otimes b\to b^*\otimes a^*$ is also studied. AMS 2000 Mathema...
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It is known that if the supports of a function f ∈ L1(Rn) and its Fourier transform have finite measure then f = 0 almost everywhere. We study generalizations of this property for certain classes of locally compact hypergroups. http://web.math.hr/glasnik/vol_36/no1_04.html
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The Haagerup norm parallel to . parallel to(h) on the tensor product A x B of two C*-algebras A and B is shown to be Banach space equivalent to either the Banach space projective norm parallel to . parallel to(gamma) or the operator space projective norm parallel to . parallel to(boolean AND) if and only if either A or B is finite dimensional or A...
Article
In this paper, we discuss the solvability of certain nth order systems arising from bi-analytic functions
Article
A function theory of complex Bi-analytic functions in a variables determined by certain second order system is developed. This has been applied to solve the Dirichlet problem of the corresponding systems.

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