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.

&nbsp;

IIT Palakkad

Let T: D(T) → H2 be a densely defined closed operator with domain D(T) ⊂ H1. We say T to be absolutely minimum attaining if for every non-zero closed subspace M of H1 with D(T) ∩ M ≠ {0}, the restriction operator T|M: D(T) ∩ M → H2 attains its minimum modulus m(T|M). That is, there exists x ∈ D(T) ∩M with ‖x‖ = 1 and ‖T (x)‖ = inf{‖T (m)‖: m ∈ D(T) ∩ M: ‖m‖ = 1}. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. One such important characterization is that an unbounded operator belongs to this class if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent. © D l, Zagreb.

Department of Mathematics

World University root node

Operators and Matrices

SPECTRAL REPRESENTATION OF ABSOLUTELY MINIMUM ATTAINING UNBOUNDED NORMAL OPERATORS

The aim of this paper is to obtain the solution of the operator equation Tx= y for an infinite band matrix T using its finite-dimensional truncations Tnxn= yn. Several verifiable conditions are given to obtain the invertibility of T. An application in connection with a stable set of sampling for functions belonging to a shift-invariant space is discussed along with an illustration. © 2022, Tusi Mathematical Research Group (TMRG).

Banach Journal of Mathematical Analysis

Solution of an infinite band matrix equation

Let A be a complex unital Banach algebra with unit 1. An element a∈ A is said to be of G1-class if ‖(z-a)-1‖=1d(z,σ(a))∀z∈C\σ(a).Here d(z, σ(a)) denotes the distance between z and the spectrum σ(a) of a. Some examples of such elements are given and also some properties are proved. It is shown that a G1-class element is a scalar multiple of the unit 1 if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if T is a G1 class operator on a Banach space X, then every isolated point of σ(T) is an eigenvalue of T. If, in addition, σ(T) is finite, then X is a direct sum of eigenspaces of T. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.

Fulltext

Journal of Analysis

G1 -class elements in a Banach algebra

In order to understand the behaviour of a square matrix or a bounded linear operator on a Banach space or more generally an element of a Banach algebra, some subsets of the complex plane are associated with such an object. Most popular among these sets is the spectrum σ(a) of an element a in a complex unital Banach algebra A with unit 1 defined as follows: σ(a):={λ∈C:λ-aisnotinvertibleinA}.Here and also in what follows, we identify λ. 1 with λ. Also quite popular is Numerical range V(a) of a. This is defined as follows: V(a):={ϕ(a):ϕisacontinuouslinearfunctionalonAsatisfying‖ϕ‖=1=ϕ(1)}.Then there are many generalizations, modifications, approximations etc. of the spectrum. Let ϵ&gt; 0 and n a nonnegative integer. These include ϵ- condition spectrum σϵ(a) , ϵ- pseudospectrum Λ ϵ(a) and (n, ϵ) - pseudospectrum Λ n,ϵ(a). These are defined as follows: σϵ(a):={λ∈C:‖λ-a‖‖(λ-a)-1‖≥1ϵ}In this and the following definitions we follow the convention : ‖ (λ- a) - 1‖ = ∞ if λ- a is not invertible. Λϵ(a):={λ∈C:‖(λ-a)-1‖≥1ϵ}Λn,ϵ(a):={λ∈C:‖(λ-a)-2n‖1/2n≥1ϵ}. In this survey article, we shall review some basic properties of these sets, relations among these sets and also discuss the effects of perturbations on these sets and the question of determining the properties of the element a from the knowledge of these sets. © 2019, Forum D'Analystes, Chennai.

Spectrum and related sets: a survey

We define and discuss properties of the class of unbounded operators which attain minimum modulus. We establish a relationship between this class and the class of norm attaining bounded operators and compare the properties of both. Also we define absolutely minimum attaining operators (for not necessarily bounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of those is that every such operator has a non trivial hyperinvariant subspace. © 2019, Forum D'Analystes, Chennai.

Absolutely minimum attaining closed operators

The paper deals with sampling and reconstruction of vector valued functions in a shift invariant space with multiple generators. Unlike the case of a shift invariant space with multiple generators in L2(R) , when the dimension of the vectors is the same as the number of generators, Z turns out to be a stable set of sampling. A sampling formula for reconstructing a function from its samples at integer points is derived and the problem of sampling on a perturbed set of integers is discussed. An illustration of sampling and reconstruction of a function in L2(R, R2) on a finite interval is given using Matlab. © 2020, Forum D'Analystes, Chennai.

Shift invariant spaces in L2(R, Cm) with m generators

Let A be a complex Banach algebra with unit. For an integer n≥ 0 and ϵ&gt; 0 , the (n, ϵ) -pseudospectrum of a∈ A is defined by Λn,ϵ(A,a):={λ∈C:(λ-a)is not invertible inAor‖(λ-a)-2n‖1/2n≥1ϵ}.Let p∈ A be a nontrivial idempotent. Then pAp= { pbp: b∈ A} is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose ap= pa. We study the relationship of Λn , ϵ(A, a) and Λn , ϵ(pAp, pa). We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the (n, ϵ) -pseudospectrum of a can be decomposed into the union of the (n, ϵ) -pseudospectra of some elements in reduced Banach algebras. © 2019, Tusi Mathematical Research Group (TMRG).

Advances in Operator Theory

Decomposition of the (n, ϵ) -pseudospectrum of an element of a Banach algebra

Let H1, H2 be complex Hilbert spaces and T be a densely defined closed linear operator from its domain D(T), a dense subspace of H1, into H2. Let N(T) denote the null space of T and R(T) denote the range of T. Recall that C(T):= D(T) ∩ N(T)⊥ is called the carrier space of T and the reduced minimum modulus γ(T) of T is defined as: γ (T) : = inf{ ‖ T(x) ‖ : x∈ C(T) , ‖ x‖ = 1 }. Further, we say that T attains its reduced minimum modulus if there exists x0 ∈ C(T) such that ∥x0∥ = 1 and ∥T(x0)∥ = γ(T). We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved.1.The operator T attains its reduced minimum modulus if and only if its Moore-Penrose inverse T† is bounded and attains its norm, that is, there exists y0 ∈ H2 such that ∥y0∥ = 1 and ∥T†∥ = ∥T†(y0)∥.2.For each ϵ &gt; 0, there exists a bounded operator S such that ∥S∥ ≤ ϵ and T + S attains its reduced minimum. © 2020, Indian National Science Academy.

Indian Journal of Pure and Applied Mathematics

Operators that Attain Reduced Minimum

In this article, we consider the condition pseudospectrum of bounded linear operators on a Banach space. The condition pseudospectrum of normal matrices and Jordan blocks are characterized and condition pseudospectral radius of the classes are found. Sub-additivity and sub-multiplicativity of the condition pseudospectral radius for commuting pairs of bounded linear operators are proved. It is shown that the condition pseudospectral radius becomes a complete algebra norm in a commutative complex unital Banach algebra. Certain examples are given to illustrate the results. The results developed are also extended to a general setting. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.

Linear and Multilinear Algebra

Condition pseudospectral radius of bounded linear operators

Let ϵ&gt; 0 , n a non-negative integer, and A a complex unital Banach algebra. Define γn: A× C→ [0 , ∞] by γn(a,z)={‖(z-a)-2n‖-1/2n,if(z-a)isinvertible0,if(z-a)is not invertible.The (n, ϵ) -pseudospectrum Λ n , ϵ(a) of an element a∈ A is defined by Λ n , ϵ(a) : = { λ∈ C: γn(a, λ) ≤ ϵ}. We show that γ is Lipschitz on A× C, γn is uniformly continuous on bounded subsets of A× C for n≥ 1 , and γn is Lipschitz on some particular domains for n≥ 1. Using these properties, we establish that the map (ϵ, a) ↦ Λ n , ϵ(a) is continuous at (ϵ, a) if and only if the level set { λ∈ C: γn(a, λ) = ϵ} does not contain any non-empty open set. In particular, this happens when a is a compact operator on a Banach space or a bounded operator on a Hilbert space or on an Lp space with 1 ≤ p≤ ∞. We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous. © 2019, Springer Nature Switzerland AG.}, author_keywords={(n, ϵ) -Pseudospectrum; Banach algebra; Pseudospectrum; Spectrum}, funding_details={2/39(2)/2015/NBHM/R&D-II/7440}, funding_text_1={The authors would like to thank the anonymous referee for his/her valuable comments that led to a considerable improvement in this paper. The first author would like to thank the Department of Atomic Energy (DAE), India (Ref No: 2/39(2)/2015/NBHM/R&D-II/7440) for financial support.}, references={Bögli, S., Siegl, P., Remarks on the convergence of pseudospectra (2014) Integral Equ. Oper. Theory, 80, pp. 303-321; Böttcher, A., Pseudospectra and singular values of large convolution operators (1994) J. Integral Equ. Appl., 6, pp. 267-301; Böttcher, A., Grudsky, S.M., Silbermann, B., Norms of inverses, spectra, and pseudospectra of large truncated Wiener–Hopf operators and Toeplitz matrices (1997) N. Y. J. Math., 3, pp. 1-31; Böttcher, A., Grudsky, S.M., Can spectral value sets of Toeplitz band matrices jump? (2002) Linear Algebra Appl., 351 (352), pp. 99-116; Davies, E.B., Pseudospectra of differential operators (2000) J. Oper. Theory, 43, pp. 243-262; Davies, E.B., A defence of mathematical pluralism (2005) Philos. Math., 13 (3), pp. 252-276; Dhara, K., Kulkarni, S.H., The (n, ϵ) -pseudospectrum of an element of a Banach algebra (2018) J. Math. Anal. Appl., 464, pp. 939-954; Dunford, N., Schwartz, J.T., (1958) Linear Operators. I. General Theory, , Interscience Publishers Ltd., London; Gallestey, E., Hinrichsen, D., Pritchard, A.J., Spectral value sets of closed linear operators (2000) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456, pp. 1397-1418; Globevnik, J., On complex strict and uniform convexity (1975) Proc. Am. Math. Soc., 47, pp. 175-178; Globevnik, J., Norm-constant analytic functions and equivalent norms (1976) Ill. J. Math., 20, pp. 503-506; Hansen, A.C., On the approximation of spectra of linear operators on Hilbert spaces (2008) J. Funct. Anal., 254, pp. 2092-2126; Hansen, A.C., Infinite-dimensional numerical linear algebra: theory and applications (2010) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466, pp. 3539-3559; Hansen, A.C., On the solvability complexity index, the n -pseudospectrum and approximations of spectra of operators (2011) J. Am. Math. Soc., 24, pp. 81-124; Hansen, A.C., Nevanlinna, O., Complexity issues in computing spectra, pseudospectra and resolvents (2017) Banach Cent. Publ., 112, pp. 171-194; Harrabi, A., Pseudospectre d’une suite d’opérateurs bornés (1998) RAIRO Modél. Math. Anal. Numér., 32, pp. 671-680; Krishnan, A., Kulkarni, S.H., Pseudospectrum of an element of a Banach algebra (2017) Oper. Matrices, 11, pp. 263-287; Reddy, S.C., Pseudospectra of Wiener–Hopf integral operators and constant-coefficient differential operators (1993) J. Integral Equ. Appl., 5, pp. 369-403; Seidel, M., On (N, ϵ) -pseudospectra of operators on Banach spaces (2012) J. Funct. Anal., 262, pp. 4916-4927; Shargorodsky, E., On the level sets of the resolvent norm of a linear operator (2008) Bull. Lond. Math. Soc., 40, pp. 493-504; Trefethen, L.N., Pseudospectra of linear operators (1997) SIAM Rev., 39, pp. 383-406; Trefethen, L.N., Embree, M., Pseudospectra gateway, , http://www.comlab.ox.ac.uk/pseudospectra, Web site; Trefethen, L.N., Embree, M., (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, , Princeton University Press, Princeton}, correspondence_address1={Dhara, K.; Department of Mathematics, India; email: kousik.dhara1@gmail.com}, publisher={Springer Basel}, issn={0378620X}, language={English}, abbrev_source_title={Integr. Equ. Oper. Theory}, document_type={Article}, source={Scopus},

Integral Equations and Operator Theory

Continuity of the (n,ϵ) -Pseudospectrum in Banach Algebras

Let H 1 , H 2 be complex Hilbert spaces and T be a densely defined closed linear operator (not necessarily bounded). It is proved that for each ε &gt; 0, there exists a bounded operator S with ‖S‖ ≤ ε such that T + S is minimum attaining. Further, if T is bounded below, that is if there exists m &gt; 0 such that ‖Tx‖ ≥ m‖x‖ for every x in the domain of T, then S can be chosen to be rank one. © 2018, Element D.O.O. All rights reserved.

On the denseness of minimum attaining operators

Let A be a complex unital Banach algebra, a∈A, n∈Z+ and ϵ&gt;0. The (n,ϵ)-pseudospectrum Λn,ϵ(a) of a is defined as Λn,ϵ(a):=σ(a)∪{λ∉σ(a):‖(λ−a)−2n‖1/2n≥[Formula presented]}. Here σ(a) denotes the spectrum of a. The usual pseudospectrum Λϵ(a) of a is a special case of this, namely Λ0,ϵ(a). It is proved that (n,ϵ)-pseudospectrum approximates the closed ϵ-neighbourhood of spectrum for large n. Further, it has been shown that (n,ϵ)-pseudospectrum has no isolated points, has a finite number of connected components and each component contains an element from σ(a). Some examples are given to illustrate these results. © 2018 Elsevier Inc.

Journal of Mathematical Analysis and Applications

The (n,ϵ)-pseudospectrum of an element of a Banach algebra

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers R, the complex numbers C or the quaternions H; (Complex version) Every complex normed division algebra is isometrically isomorphic to C. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues. © 2017 Elsevier GmbH

Expositiones Mathematicae

Gelfand–Mazur Theorems in normed algebras: A survey

Let A be a Banach algebra with identity 1 and p ∈ A be a nontrivial idempotent. Then q = 1-p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. For a ∈ A and ε &gt; 0, we examine the relationship between the ε-pseudospectrum Λε(A, a) of a ∈ A, and ε-pseudospectra of pap ∈ pAp and qaq 2 qAq. We also extend this study by considering a finite number of idempotents p1, · · ·, pn, as well as an arbitrary family of idempotents satisfying certain conditions. © 2016 by the Tusi Mathematical Research Group.

Pseudospectra of elements of reduced banach algebras

The ε-pseudospectrum Λε (a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a. Suppose for some ε &gt; 0 and a,b ∈ A, Λε (ax) = Λε (bx) ∀x ∈ A. It is shown that a = b if: (i) a is invertible. (ii) a is Hermitian idempotent. (iii) a is the product of a Hermitian idempotent and an invertible element. (iv) A is semisimple and a is the product of an idempotent and an invertible element. (v) A = B(X) for a Banach space X. (vi) A is a C∗ -algebra. (vii) A is a commutative semisimple Banach algebra. © ELEMENT, Zagreb.

Pseudospectrum of an element of a Banach algebra

The Nullity theorem says that certain pairs of submatrices of a square invertible matrix and its inverse (known as complementary submatrices) have the same nullity. Though this theorem has been around for quite some time and also has found several applications, some how it is not that widely known. We give a brief account of the Nullity Theorem, consider its generalization to infinite dimensional spaces, called the Null Space Theorem and discuss some applications. © Springer International Publishing Switzerland 2015.

Springer Proceedings in Mathematics and Statistics

The nullity theorem, its generalization and applications

In this paper we use Banach algebra techniques to study the spectrum, pseudospectrum and condition spectrum of a block Laurent operator with continuous symbol and a lower triangular block Toeplitz operator with continuous symbol. (1) Let L be a block Laurent operator with a continuous symbol f. Regarding f as an element of the Banach algebra of all continuous matrix valued functions defined on the unit circle Γ, we show that the spectrum σ(L) of L coincides with the spectrum σ(f) of f. It is also shown that the spectrum σ(f) can be expressed as a union of the spectra of matrices f(x). Thus Similar results are proved about pseudospectrum for ε &gt; 0 and condition spectrum σε(L) = σε(f) for 0 &lt; ε &lt; 1. (2) Let T be an upper or lower triangular block Toeplitz operator with continuous symbol f. Then f is a continuous matrix valued function defined on the closed unit disc and f is analytic in the open unit disc. It is proved that a similar description can be given about the spectrum σ(T), pseudospectrum Λε(T) for ε &gt; 0 and condition spectrum σε(T) for &lt; ε &lt; 1. These results are illustrated with examples and pictures using Matlab.

Annals of Functional Analysis

Banach algebra techniques to compute spectra, pseudospectra and condition spectra of some block operators with continuous symbols

The following results are proved. The Null Space TheoremLet X,Y be vector spaces, P∈L(X),Q∈L(Y) be projections and T∈L(X,Y) be invertible. (The restriction of QTP to R(P) can be viewed as a linear operator from R(P) to R(Q). This is called a section of T by P and Q and will be denoted by TP,Q.) Then there is a linear bijection between the null space of the section TP,Q of T and the null space of its complementary section TIY-Q,IX-P-1 of T-1. Let X be a Banach space with a Schauder basis A={a1,a2,⋯}. Let T be a bounded (continuous) linear operator on X. Suppose the matrix of T with respect to A is tridiagonal. If T is invertible, then every submatrix of the matrix of T-1 with respect to A that lies on or above the main diagonal (or on or below the main diagonal) is of rank ≤1. © 2015 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

The null space theorem

The following result is proved: Let A be a commutative real Banach algebra with unit 1. Let G denote the group of invertible elements of A and let G1 be the connected component of G containing 1. If the quotient group G/G1 contains an element of finite order other than G1, then the order of such an element must be 2. If the group G/G1 is of finite order, then its order must be 2n for some nonnegative integer n. © 2014 University of Houston.

Houston Journal of Mathematics

The group of invertible elements of a real Banach algebra

For 0 &lt; ε &lt; 1, the ε-condition spectrum of an element a in a complex unital Banach algebra A is defined as, (Formula Presented). This is a generalization of the idea of spectrum introduced in [5]. This is expected to be useful in dealing with operator equations. In this paper we prove a mapping theorem for condition spectrum, extending an earlier result in [5]. Let f be an analytic function in an open set Ω containing σε(a). We study the relations between the sets σε(f(a)) and f(σε (a)). In general these two sets are different. We define functions φ(ε), ψ (ε) (that take small values for small values of ε) and prove that f(σε(a)) ⊆ σφ(ε)(f(a)) and σε(f(a)) ⊆ f(σ ψ(ε)(a)). The classical Spectral Mapping Theorem is shown as a special case of this result. We give estimates for these functions in some special cases and finally illustrate the results by numerical computations. © 2014 Springer Basel.