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

We derive estimates for approximation numbers of bounded linear operators between normed linear spaces. As special cases of our general results, approximation numbers of some weighted shift operators on p and those of isometries andprojections of norm 1 are found. In the case of finite-rank operators, we obtain estimatesfor the smallest nonzero approximation number in terms of their generalizedinverses. Also, we prove some results regarding the relation between approximationnumbers and the closedness of the range of an operator.We recall that the closednessof the range is a necessary condition for the boundedness of a generalized inverse.We give examples illustrating the results and also show that certain inequalities neednot hold. © Springer India 2013.

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Department of Mathematics

Combinatorial Matrix Theory and Generalized Inverses of Matrices

Generalized inverses and approximation numbers

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A, B be complex unital Banach algebras and  > 0. Let : A  B be an -pseudospectrum preserving linear onto map. Then  preserves spectrum. If A and B are uniform algebras, then,  is an isometric isomorphism. (2) Let A, B be uniform algebras and 0 <  < 1. Let : A  B be an -condition spectrum preserving linear map. Then  is an -almost multiplicative map, where ,  tend to zero simultaneously.

Banach Journal of Mathematical Analysis

Linear maps preserving pseudospectrum and condition spectrum

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T ↦ T† is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric.

The carrier graph topology

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T → T† is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric. © 2010 Mathematics Subject Classification.

Aim of this short note is to obtain a formula for the gap between two densely defined unbounded closed operators. It is interesting to note that the formula is very similar to the corresponding formula for the gap between two bounded operators. © 2010 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

A formula for gap between two closed operators

Let A be a complex commutative Banach algebra with unit 1 and δ &gt; 0. A linear map φ: A →C is said to be δ-almost multiplicative if |φ(ab) - φ (a) φ (b)| ≤ δ||a|| ||b|| for all a, b ε A. Let 0 &lt; e &lt; 1. The e-condition spectrum of an element a in A is defined by δe.(a) := {λ ε C:||λ-a||||λ- a||-1 ||≥ 1/e} with the convention that ||λ- a|| ||(λ - a)-1|| = ∞ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If φ(1) = 1 and φ is δ-almost multiplicative, then φ(a) ε δe(a) for all a in A.then (2) If φis lenear and φ(a) εδe(a) for all a in A ,then φ-is δ almost multiplicative for some δ. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane- Żelazko theorem.

Studia Mathematica

Almost multiplicative functions on commutative Banach algebras

In this article we give a characterization of the convergence of projection methods which are useful for approximating the Moore-Penrose inverse of a closed densely defined operator between Hilbert spaces. We illustrate the main theorem with an example. Also a procedure for constructing the admissible sequence of projections is discussed. © 2010 The Indian National Science Academy.

Indian Journal of Pure and Applied Mathematics

Projection methods for computing Moore-Penrose inverses of unbounded operators

In [1], Böttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H,{ej}j=1∞ is an orthonormal basis of H and Pn is the orthogonal projection onto the span of {ej}j=1n, then for each k ∈ Ndbl;, the sequence {sk(pnTPn)} converges to sk(T), where for a bounded operator A on H, sk(A) denotes the kth approximation number of A, that is, sk(A) is the distance from A to the set of all bounded linear operators of rank at most k - 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {Pn} and {Qn} are sequences of bounded linear operators on X and Y, respectively, such that for all {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 for all n ∈ ℕ and {QnTPn} converges to T under the weak operator topology, then {sk(QnTPn)} converges to sk(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of sk(QnTPn) to sk(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {Pn} and {Qn} on X and Y respectively such that (i) QnTPn is compact, (ii) {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 and (iii) {QnTPn} converges to T in the weak operator topology, then {sk(QnTPn)} converges to sk(T) if and only if sk(T) = sk(T′). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces. © 2009 Birkhäuser Verlag Basel/Switzerland.

Integral Equations and Operator Theory

Approximation numbers of operators on normed linear spaces

Let H1,H2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H1 and taking values in H2. In this article we prove the following results: (i) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T*T) of T*T, In addition, if H1 = H2 and T is self-adjoint, then (ii) inf {{double pipe}Tx{double pipe}:x∈D(T)∩N(T)⊥{double pipe}x{double pipe} = 1} = inf{|λ|: 0 ≠ λ ∈ σ(T)}, (iii) Every isolated spectral value of T is an eigenvalue of T (iv) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T) of T (v) σ(T) bounded implies T is bounded. We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

Some properties of unbounded operators with closed range

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.

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.

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.

A characterization of closed range operators

Three Gelfand-Mazur type theorems are proved. One of these provides a C*-property analogue of Zalar's recent generalizations of the EroelichIngelstam-Smiley Theorems concerning unital multiplication in Hilbert spaces; the second illustrates that the assumption in Kaplansky's version of the Gelfand-Mazur Theorem can be weakened in the presence of a C*-norm; whereas the third provides a real analogue of a result due to Srinivasan. © 1998 American Mathematical Society.