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Full-Text Articles in Physical Sciences and Mathematics
Commuting Isometries And Invariant Subspaces In Several Variables., Sankar T. R. Dr.
Commuting Isometries And Invariant Subspaces In Several Variables., Sankar T. R. Dr.
Doctoral Theses
A very general and fundamental problem in the theory of bounded linear operators on Hilbert spaces is to find invariants and representations of commuting families of isometries.In the case of single isometries this question has a complete and explicit answer: If V is an isometry on a Hilbert space â„‹, then there exists a Hilbert space Hu and a unitary operator U on â„‹u such that V on â„‹u and[ S ⊗ IW 0 0 U] ∈ B((l 2 (ℤ+) ⊗ W) ⊕ â„‹u),are unitarily equivalent, whereW = ker V∗ ,is the wandering subspace for V and S is the …
Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.
Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.
Doctoral Theses
No abstract provided.
Quantum Markov Maps: Structureand Asymptotics., Vijaya Kumar U. Dr.
Quantum Markov Maps: Structureand Asymptotics., Vijaya Kumar U. Dr.
Doctoral Theses
No abstract provided.
A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.
A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.
Doctoral Theses
In analytic function theory, the study of multiplication and composition operators has a rich structure for various analytic function spaces of the unit disk D = {z ∈ C : |z| < 1} such as the Hardy spaces Hp, the Bergman spaces Ap and the Bloch space B. This theory connects the operator theoretic properties such as boundedness, compactness, spectrum, invertibility, isometry with that of the function theoretic properties of the inducing map (symbol) such as bijectivity, boundary behaviour and vise versa In Chapter 2, we define discrete analogue of generalized Hardy spaces (Tp) and their separable subspaces (Tp,0) on a homogenous rooted tree and study some of their properties such as completeness, inclusion relations with other spaces, separability and growth estimate for functions in these spaces and their consequences. In Chapter 3, we obtain equivalent conditions for multiplication operators Mψ on Tp and Tp,0 to be bounded and compact. Furthermore, we discuss point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. In Chapter 4, we give an equivalent conditions for the composition operator Cφ to be bounded on Tp and on Tp,0 spaces and compute their operator norms. We have considered the composition operators induced by special symbols such as univalent and multivalent maps and automorphism of a homogenous tree. We also characterize invertible composition operators and isometric composition operators on Tp and on Tp,0 spaces. Also, we discuss the compactness of Cφ on Tp spaces and finally we prove that there are no compact composition operators on Tp,0 spaces. In Chapter 5, we consider the composition operators on the Hardy-Dirichlet space H2, the space of Dirichlet series with square summable coefficients. By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on H2 , for the affine-like inducing symbol ϕ(s) = c1 + cqq −s , where q ≥ 2 is a fixed integer. We also give an estimate for approximation numbers of a composition operators in our H2 setting. In Chapter 6, we study the weighted composition operators preserving the class Pα. Some of its consequences and examples of certain special cases are presented. Furthermore, we discuss about the fixed points of weighted composition operators.
On Free-Type Rigid C*-Tensor Categories And Their Annular Representations., B. Madhav Reddy Dr.
On Free-Type Rigid C*-Tensor Categories And Their Annular Representations., B. Madhav Reddy Dr.
Doctoral Theses
No abstract provided.
Infinite Mode Quantum Gaussian States., Tiju Cherian John Dr.
Infinite Mode Quantum Gaussian States., Tiju Cherian John Dr.
Doctoral Theses
No abstract provided.
Nonparametric Methods For Data In Infinite Dimensional Space., Anirvan Chakraborty Dr.
Nonparametric Methods For Data In Infinite Dimensional Space., Anirvan Chakraborty Dr.
Doctoral Theses
For univariate as well as finite dimensional multivariate data, there is an extensive literature on nonparametric statistical methods. One of the reasons for the popularity of nonparametric methods is that it is often difficult to justify the assumptions (e.g., Gaussian distribution of the data) made in the models used in parametric methods. Nonparametric procedures use more flexible models, which involve less assumptions. So, they are more robust against possible departures from the model assumptions, and are applicable to a wide variety of data. Nonparametric methods outperform their parametric competitors in many situations, where the assumptions required for the parametric methods …
Some Problems In Differential And Subdifferential Calculus Of Matrices., Priyanka Grover Dr.
Some Problems In Differential And Subdifferential Calculus Of Matrices., Priyanka Grover Dr.
Doctoral Theses
A central problem in many subjects like matrix analysis, perturbation theory, numerical analysis and physics is to study the effect of small changes in a matrix A on a function f(A). Among much studied functions on the space of matrices are trace, determinant, permanent, eigenvalues, norms. These are real or complex valued functions. In addition, there are some interesting functions that are matrix valued. For example, the (matrix) absolute value, tensor power, antisymmetric tensor power, symmetric tensor power.When a function is differentiable, one of the ways to study the above problem is by using the derivative of f at A, …
Bures Distance For Completely Positive Maps And Cp-H-Extendable Maps Between Hilbert C*- Modules., Sumesh K Dr.
Bures Distance For Completely Positive Maps And Cp-H-Extendable Maps Between Hilbert C*- Modules., Sumesh K Dr.
Doctoral Theses
Completely positive (CP-) maps are special kinds of positivity preserving maps on C ∗ -algebras. W.F. Stinespring [Sti55] obtained a structure theorem for CP-maps showing that they are closely connected with ∗-homomorphisms. W. Arveson and other operator algebraists quickly realized the importance of these maps. Presently the role of the theory of CP-maps in our understanding of C ∗ -algebras and von Neumann algebras is well recognised. It has been argued by physicists that CPmaps are physically more meaningful than just positive maps due to their stability under ampliations. From quantum probabilistic point of view CP-maps are quantum analogues of …
Geometric Invariants For A Class Of Semi-Fredholm Hilbert Modules., Shibananda Biswas Dr.
Geometric Invariants For A Class Of Semi-Fredholm Hilbert Modules., Shibananda Biswas Dr.
Doctoral Theses
One of the basic problem in the study of a Hilbert module H over the ring of polynomials C[z] := C[z1, . . . , zm] is to find unitary invariants (cf. [15,7]) for H. It is not always possible to find invariants that are complete and yet easy to compute. There are very few instances where a set of complete invariants have been identified. Examples are Hilbert modules over continuous functions (spectral theory of normal operator), contractive modules over the disc algebra (model theory for contractive operator) and Hilbert modules in the class Bn for a bounded domain C …
Quantum Stochastic Flows: Trotter Product Formula, Dilations And Quantum Brownian Motion., Biswarup Das Dr.
Quantum Stochastic Flows: Trotter Product Formula, Dilations And Quantum Brownian Motion., Biswarup Das Dr.
Doctoral Theses
Motivated by the major role played by probabilistic models in many areas of science, several quantum (i.e. non-commutative) generalizations of classical probability have been attempted by a number of mathematicians. The pioneering works of K.R. Parthasarathy, L. Accardi, R.L. Hudson, P.A. Meyer and others led to the development of one such non-commutative model called ‘quantum probability’ which has a very rich theory of quantum stochastic calculus a la Hudson and Parthasarathy. Within the framework of quantum stochastic calculus, the ‘grand design’ that engages us is the canonical construction and study of ∗-homomorphic flows (jt)t≥0 on a given C ∗ or …
Homogeneous Operators In The Cowen-Douglas Class., Subrata Shyam Roy Dr.
Homogeneous Operators In The Cowen-Douglas Class., Subrata Shyam Roy Dr.
Doctoral Theses
Although, we have used techniques developed in the paper of Cowen-Douglas [18, 20], a systematic account of Hilbert space operators using a variety of tools from several different areas of mathematics is given in the book [26]. This book provides, what the authors call, a sheaf model for a large class of commuting Hilbert space operators. It is likely that these ideas will play a significant role in the future development of the topics discussed here.
Uncertainty Principles On Nilpotent Lie Groups., Sanjay Parui Dr.
Uncertainty Principles On Nilpotent Lie Groups., Sanjay Parui Dr.
Doctoral Theses
No abstract provided.
Quantum Stochastic Dilation Of A Class Of Quantum Dynamical Semigroups And Quantum Random Walks., Lingaraj Sahu Dr.
Quantum Stochastic Dilation Of A Class Of Quantum Dynamical Semigroups And Quantum Random Walks., Lingaraj Sahu Dr.
Doctoral Theses
No abstract provided.
Spectral Triples And Metric Aspects Of Geometry On Some Noncommutative Spaces., Partha Sarathi Chakraborty Dr.
Spectral Triples And Metric Aspects Of Geometry On Some Noncommutative Spaces., Partha Sarathi Chakraborty Dr.
Doctoral Theses
Quantization of mathematical theories is now more than half a century old idea in mathe- matics. It goes back to Gelfand-Naimarks seminal paper [37] in 1943. As the name suggests noncommutative geometry is the quantization" of differential geometry. It is the study of noncommutative algebras as if they were algebras of functions on spaces like the commuta- tive algebras associated to affine algebraic varieties, smooth manifolds, topological spaces. One can trace its roots in the Gelfand-Naimark theorems (1943, 37]). In modern terminol- ogy their theorem says there is an antiequivalence between the category of (locally) compact Hausdorff spaces and (proper, …
Quantum Stochastic Dilation Of Completely Positive Semigroups And Flows., Debashish Goswami Dr.
Quantum Stochastic Dilation Of Completely Positive Semigroups And Flows., Debashish Goswami Dr.
Doctoral Theses
The central theme of the present thesia is quantum stochastic dilation af semigroupe of completely panitive mapa on operator algebran. It is the sim of all mathemati- cal, or even all scientific theorics, to understand a given class of objects through a tanonical and simpler subclass of it. For example, abstract C"-algebras are studied through their conerete realisation as elgebra of operators, contractions on a Hilbert space by unitaries. Hilbert modules by the factorissble ones, to mention anly a few. In most af these caes, a general object of the relavant class is sociated with a canonical candidate of the …
Time-Space Harmonic Polynomials For Stochastic Processes., Arindam Sengupta Dr.
Time-Space Harmonic Polynomials For Stochastic Processes., Arindam Sengupta Dr.
Doctoral Theses
The sequence of polynomials of a single variable known as the Hermite polynomialshala) = ), k21, (-1)* ha(z) =has many close links with the Normal distribution. Their association goes very doep, and extends to several connections bet ween the two-variable Hermite polynomialsHll, 2) = the(z/t), . k21.and the prime example of Gaussian processes, that is Brownian motion, as well. Much of this connection stems from what we term the time-space harmonic property of these polynomials for the Brownian motion process. An exact definition of this property follows later. A natural question that arises is, for stochastic processes in general, when …
Wiener Tauberian Theorems On Semisimple Lie Group., Rudra Pada Sarkar Dr.
Wiener Tauberian Theorems On Semisimple Lie Group., Rudra Pada Sarkar Dr.
Doctoral Theses
In their celebrated study of Harmonic analysis on semi-simple Lie groups Ehrenpreis and Mautner [E-M] noticed that the analogue of the claasical Wiener Tauberian theorem resting on the unitary dual does not hold for semisimple Lle groups. A simple proof of this fact due to M. Duflo appears in (H). Ehrenpreis and Mautner went on in (E-M] to formulate the problem on the commutative Banach algebra of the SO2(R)-biinvariant functions in L1(SL2(R))1, and obtained two different versions of the theorem involving, this time, the dual of the Banach algebra which includes, beside the unitary dual of G, a part of …
Fourier Transforms Of Very Rapidly Decreasing Functions On Certain Lie Groups., M. Sundari Dr.
Fourier Transforms Of Very Rapidly Decreasing Functions On Certain Lie Groups., M. Sundari Dr.
Doctoral Theses
Recall that for a function f ϵ L1(Rn ), its Fourier transform fÌ‚ is definedby: fÌ‚ (ƹ) = ʃ Rnf(x)ei(ƹ,x)dx ( 0.1.1)where (.,.) denotes the standard inner product on Rn and dr the Lebesgue measure on Rn. A celebrated theorem of L. Schwartz asserts that a function f on Rn is rapidly decreasing (or in the Schwartz class ) if and only if its Fourier transform is rapidly decreasing . In sharp contrast to Schwartz s theorem, is a result due to Hardy ([18) which says that ʃ and fÌ‚ cannot both be very rapidly decreasing . More precisely, if …
Some Problems In Joint Spectral Theory., Tirthankar Bhattacharya Dr.
Some Problems In Joint Spectral Theory., Tirthankar Bhattacharya Dr.
Doctoral Theses
No abstract provided.
Markov Dilation Of Nonconservative Quantum Dynamical Semigroups And Quantum Boundary Theory., B. V. Rajarama Bhat Dr.
Markov Dilation Of Nonconservative Quantum Dynamical Semigroups And Quantum Boundary Theory., B. V. Rajarama Bhat Dr.
Doctoral Theses
In classical probability theory, based on Kolmogorov consistency theorem, one can associate a Markov process to any one parameter semigroup of stochastic matrices or transition probability operators. It is indeed the foundation for the theory of Markov processes. Here a quantum version of this theorem has been established. This effectively answers some of the questions raised by P. A. Meyer in his book (see page 220 of (Me).It is widely agreed upon that irreversible dynamics in the quantum setting is de- scribed by contractive semigroups of completely positive maps on C" algebras ([Kr). (AL]). In other words these semigroups, known …
On Markov Processes Charecterised Via Martingale Problem., Abhay G. Bhatt Dr.
On Markov Processes Charecterised Via Martingale Problem., Abhay G. Bhatt Dr.
Doctoral Theses
Martingale approach to the study of finite dimensional diffusions was initiated by Stroock-Varadhan, who coined the term martingale problem. Their success led to a similar approach being used to study Markov processes occuring in other areas such as infinite particle systems, branching processes, genetic models, density dependent population processes, random evolutions etc.Suppose X is a Markov process corresponding to a semigroup (T)e20 with generator L. Then all the information about X is contained in L. We also have thatMf(t) := f(X(t)) – ∫t0 Lf(X(s))dsis a martingale for every f ∈ D(L). i.e. X is a solution to the martingale problem …
Some Spectral Properties Of Three And Four Body Scrodinger Operators By The Methods Of Time Dependent Scattering Theory., M. Krishna Dr.
Some Spectral Properties Of Three And Four Body Scrodinger Operators By The Methods Of Time Dependent Scattering Theory., M. Krishna Dr.
Doctoral Theses
In the present work we shall deal with the asymptotic completeness problem in three and four (Quantum Mechanical) particle scattering. This thesis is divided into three chapters. In the first chapter we give an introduction to Scattering Phencmenon and give a description of the N-particle completeness problem.Then we collect some results preliminary to the later chapters and some results that would complete a discussion of the problem.The second chapter consists of some technical results and a reduction of the asymptotic completeness problem in N-particlerscattering via time dependent methods. The last chapter has two sections. The first section deals with verifying …
Spectral And Scattering Theory For Schrodinger Operator With A Class Of Momentum Dependent Long Range Potentials., P. L. Muthuramalingam Dr.
Spectral And Scattering Theory For Schrodinger Operator With A Class Of Momentum Dependent Long Range Potentials., P. L. Muthuramalingam Dr.
Doctoral Theses
No abstract provided.
System Of Imprimitivity For Ergodic Actions Of Locally Compact Abelian Groups., Joseph Mathew Dr.
System Of Imprimitivity For Ergodic Actions Of Locally Compact Abelian Groups., Joseph Mathew Dr.
Doctoral Theses
Mackeys theorem on inducud representation gives all the systems of imprimitivity for a locally compact second countable group G, acting in a separable Hilbert space and based on a transitive G-space X. These systems of imprimitivity are obtained from the unitary representationg of the closed subgroup H of G defining the transitive G-space X. The main tool for this study is a class of functions called cocycles. Maçkey showed that the (G, x) systems of imprimitivity are connected in a one-one way to unitary operator valued cocycles on GXX: This cocycle in turn gives a representation of the closed subgroup …
Invariant Subspaces Of Vector Valued Function Spaces On Bohr Groups., Somesh Chandra Bagchi Dr.
Invariant Subspaces Of Vector Valued Function Spaces On Bohr Groups., Somesh Chandra Bagchi Dr.
Doctoral Theses
The theory of invariant subspaces of various Cunclion-spaces of complox-valued and vector-valued functions on the circle group 13 well known through the 'Loctures on Invariant Subspaces' by Helson ([4]), Replacin; Line circle croup by a Bohr group B (that is, a compact ahelian group whose dual is a subgroup of tho ronl. line R, dense in the topology of R), Helson and Lowdenalager initiated the study of invariant subspaces of L2,(B) in (6]. They discovered that after suitable normalisation, the simply invariant subspaces of L2(B) arcinonc-to-one correspondence with a certain class of functions on R x B, which are called …
Contribution To The Ergodic Theory., S. Natarajan Dr.
Contribution To The Ergodic Theory., S. Natarajan Dr.
Doctoral Theses
Ergodic theory is chiefly concerned with the study of transformations on a measure space which preserve the measure. Interesting classes of such transformations are the classes of ergodic, weakly mixing and mixing transformations. The bulk of this thesis is devoted to the study of a closely related family of transformations called the we akly stable transformations; these are more general than the weakly mäxing transformations.This thesis is divided into three parts. Part I contains preliminary ideas, notations and some results on the invertibility and continuity properties of transition functions (Chapters 1 and 2). In Chapter 3 we collect known results …