Namita Das

Bio: Namita Das is an academic researcher from Utkal University. The author has contributed to research in topics: Bergman space & Bergman kernel. The author has an hindex of 7, co-authored 30 publications receiving 131 citations. Previous affiliations of Namita Das include Sambalpur University & University of Leeds.

Numerical radius inequalities for operator matrices

Abstract: ‎Several numerical radius inequalities for operator matrices are‎ ‎proved by generalizing earlier inequalities‎. ‎In particular‎, ‎the‎ ‎following inequalities are obtained‎: ‎if $n$ is even‎, ‎‎ \[2w(T) \leq \max\{\| A_1 \|‎, ‎\| A_2 \|,\ldots‎, ‎\| A_n \|\}+\frac{1}{2}\displaystyle\sum_{k=0}^{n-1} \|‎~ ‎|A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|,\] ‎and if $n$ is odd‎,‎‎ \[2w(T) \leq \max\{\| A_1 \|,\| A_2 \|,\ldots,\| A_n \|\}‎+ ‎w\bigg(\widetilde{A}_{(\frac{n+1}{2})t}\bigg)‎+ ‎\frac{1}{2}\displaystyle\sum_{k=0}^{n-1} \|‎~ ‎|A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|,\] ‎for all $t\in [0‎, ‎1]$‎, ‎$ A_i$'s are bounded linear operators on the‎ ‎Hilbert space $\mathcal{H}$‎, ‎and $T$ is off diagonal matrix with entries ‎$‎A_1, \cdots, A_n‎$‎.‎

Little Hankel operators on the half plane

Abstract: In this paper we consider a class of weighted integral operators onL2 (0, ∞) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.

New inequalities similar to Hardy-Hilbert's inequality

Abstract: In this paper, we establish a new inequality similar to Hardy-Hilbert's inequality. As applications, some particular results and the equivalent form are derived. The integral analogues of the main results are also given.

Berezin number and numerical radius inequalities for operators on Hilbert spaces

Abstract: In this article, we establish several Berezin number inequalities for the quadratic weighted operator geometric mean of operators on reproducing kernel Hilbert spaces and numerical radius inequalities for operators having similar positive parts. We also provide a concluding section which revisits the connection between the numerical radius and generalized inverses, and may spark new problems for future research interest.

A Course In Functional Analysis

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Function theory in several complex variables

Abstract: Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Abstract: Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index