Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

Department of Mathematics

Let T be a closed linear operator defined on a dense subspace D(T) of a separable complex Hilbert space H and suppose that T has a bounded inverse. We characterize the convergence of projection methods for solving the operator equation Tx = y. We discuss the stability of convergence of projection methods under bounded perturbations. Also we discuss convergence of modified projection methods. We give examples to illustrate our results.

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Indian Journal of Pure and Applied Mathematics

Projection methods for inversion of unbounded operators

A relatively elementary proof of a characterization of polynomials

We give an elementary proof of a result which characterizes onto *-isomorphisms of the algebra BL(H) of all the bounded linear operators on a Hubert space H. A known proof of this result (Arveson, 1976) relies on the theory of irreducible representations of C*-algebras, whereas the proof given by us is based on elementary properties of operators on a Hilbert space which can be found in any introductory text on Functional Analysis. © 2005 American Mathematical Society.

Fulltext

Proceedings of the American Mathematical Society

An elementary proof for a characterization of *-isomorphisms

Let H be a separable Hilbert space with an orthonormal basis {en/n∈N}, T be a bounded tridiagonal operator on H and Tn be its truncation on span ({e1, e2, ... , en}). We study the operator equation Tx = y through its finite dimensional truncations T nxn = yn. It is shown that if {∥Tn-1en∥} and {∥Tn*-1en∥} are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence {Tn-1}. We also give sufficient conditions for the boundedness of {∥Tn-1en∥} and {∥Tn*-1en∥} in terms of the entries of the matrix of T. © 2005 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

Solution of a tridiagonal operator equation

We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ: A → ℂ be a linear map such that φ(1) = 1 and (φ(a))2 + (φ(b))2 ≠ 0 for all a, b in A satisfying ab = ba and a2 + b2 is invertible. Then φ(ab) = φ(a)φ(b) for all a, b in A. Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum. Copyright © 2005 Hindawi Publishing Corporation.

International Journal of Mathematics and Mathematical Sciences

Gleason-Kahane-Żelazko theorem for spectrally bounded algebra

American Mathematical Monthly

A Very Simple and Elementary Proof of a Theorem of Ingelstam

For a Banach algebra A, rad(A) denotes the radical of A and for a ∈ A, r(a) denotes the spectral radius of A. The following theorem is proved: Let A and B be real commutative Banach algebras, each with the unit 1. Let T: A → B be a linear map satisfying T(1) = 1, T is onto and r(T(a)) = r(a) for all a ∈ A. Then T is a homomorphism modulo rad(B). (This means that T(ab) - T(a)T(b) ∈ rad(B) for all a, b ∈ A). If in addition, B is semisimple, then T is a homomorphism. Further, if A is also semisimple, then T is an isomorphism. As a consequence, we get the following: Let X, Y be compact Hausdorff spaces, let C(X), C(Y ) denote the Banach algebras of all complex valued continuous functions on X, Y respectively each with the supremum norm. Let A, B be uniformly closed real subalgebras of C(X), C(Y ) respectively, each containing the constant function 1. Let T: A → B be a linear onto map. Then T is an isometry and T(1) = 1 if and only if T is an algebra isomorphism. These results generalize the classical theorems of Banach-Stone and Nagasawa.

Real Analysis Exchange

Spectral isometries of real commutative Banach algebras

It is shown that the almost Mathieu operators of the type Ten = en-1 + λsin(2nr)en + en+1 where λ is real and r is a rational multiple of π and {en : n = 1,2,3,...}, an orthonormal basis for a Hilbert space, is not invertible. © 2000 American Mathematical Society.

Non-invertibility of certain almost mathieu operators

A quantitative version of the Arzela-Ascoli theorem is proved. This version implies that a closed and bounded subset of C(X) is nearly compact, if and only if, it is nearly equicontinuous. © 2000 Taylor & Francis Group, LLC.

International Journal of Mathematical Education in Science and Technology

Arzela-Ascoli theorem is stable

We prove that if T : X → Y is a bounded linear operator between Hubert spaces X and Y, then the range of T is closed in Y if and only if 0 is not an accumulation point of the spectrum of T*T. We supply three proofs of this result as applications of different concepts in basic operator theory.