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IIT Hyderabad

In this article we present the polar decomposition of a bounded linear operator defined on an infinite-dimensional complex Hilbert space with respect to a conjugation. As a result, the polar decomposition for C-symmetric, C-skew-symmetric, and C-normal operators are obtained. As an another consequence the Godič –Lucenko theorem is obtained. We give two applications of the polar decomposition theorem. First, we obtain the polar decomposition of bounded antilinear operators and discuss various cases. Second, we derive an inequality for the bounded linear operator based on the Hölder-McCarthy’s inequality and derive an inequality for C-normal operators. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.

Department of Mathematics

World University root node

Monatshefte fur Mathematik

On the C-polar decomposition of operators and applications

In this article, first, we study the spectral properties of operators which attain their norm on every reducing subspace and then we study the structure of normal and quasinormal operators in this class. At the end we give a description of paranormal operators whose adjoint is also paranormal. This gives a direct proof of the fact that an operator is normal if and only if the operator and its adjoint are paranormal. © 2022, Tusi Mathematical Research Group (TMRG).

Annals of Functional Analysis

On operators which attain their norm on every reducing subspace

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if (Formula presented.). In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem. © 2022 Informa UK Limited, trading as Taylor &amp; Francis Group.

Linear and Multilinear Algebra

A representation of compact C-normal operators

Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)&gt;ess inf|ψ|, which improves a result from [15]. © 2022 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators

We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cφ on the Segal-Bargmann space H(ϵ), where φ(z) - Az + b, A is a bounded linear operator on ϵ, b ∈ ϵ with |A| ≤ 1 and A∗b belongs to the range of (I - A∗A)1/2. Specifically, under some conditions on the symbol φ we show that if Cφ is cyclic then A∗ is cyclic but the converse need not be true. We also show that if Cφ∗ is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space H(ϵ) for certain class of symbols φ. © 2022 G. Ramesh et al., published by De Gruyter.

Fulltext

Concrete Operators

Cyclic Composition operators on Segal-Bargmann space

Let (Formula presented.) and (Formula presented.) be complex Hilbert spaces and (Formula presented.) be a bounded linear operator. We say (Formula presented.) is norm attaining if there exists (Formula presented.) with (Formula presented.) such that (Formula presented.). If for every non-zero closed subspace (Formula presented.) of (Formula presented.), the restriction (Formula presented.) is norm attaining, then (Formula presented.) is called an absolutely norm attaining operator or (Formula presented.) -operator. If we replace the norm of the operator by the minimum modulus (Formula presented.) in the above definitions, then (Formula presented.) is called a minimum attaining and an absolutely minimum attaining operator or (Formula presented.) -operator, respectively. In this article, we discuss the operator norm closure of (Formula presented.) -operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for (Formula presented.) -operators and prove that the closure of (Formula presented.) -operators is the same as the closure of (Formula presented.) -operators. Consequently, we prove similar results for operators in the norm closure of (Formula presented.) -operators. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

On the closure of absolutely norm attaining operators

In this article, we study the Bishop-Phelps-Bollobás type theorem for minimum attaining operators. More explicitly, if we consider a bounded linear operator T on a Hilbert space H and a unit vector x0 ∈ H such that ‖Tx0‖ is very close to the minimum modulus of T, then T and x0 are simultaneously approximated by a minimum attaining operator S on H and a unit vector y ∈ H for which ‖Sy‖ is equal to the minimum modulus of S . Further, we extend this result to a more general class of densely defined closed operators (need not be bounded) in Hilbert space. As a consequence, we get the denseness of the set of minimum attaining operators in the class of densely defined closed operators with respect to the gap metric. © ELEMENT, Zagreb.

Operators and Matrices

A bishop–phelps–bollobÁs type property for minimum attaining operators

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizations are useful in checking C-normality of an operator. We illustrate these results with examples. © 2021 Informa UK Limited, trading as Taylor &amp; Francis Group.

Cartesian decomposition of C-normal operators

A bounded linear operator T: H1 → H2, where H1, H2 are Hilbert spaces, is said to be norm attaining if there exists a unit vector x ∈ H1 such that kT xk = kT k and absolutely norm attaining (or AN -operator) if T |M: M → H2 is norm attaining for every closed subspace M of H1. We prove a structure theorem for positive operators in β(H):= {T ∈ B(H): T |M: M → M is norm attaining for all M ∈ RT }, where RT is the set of all reducing subspaces of T . We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class. © 2021 University of Szeged. All rights reserved.

Acta Scientiarum Mathematicarum

On a subclass of norm attaining operators

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.

Journal of Analysis

Absolutely minimum attaining closed operators

In this article, we characterize absolutely norm attaining normal operators in terms of the essential spectrum. Later we prove a structure theorem for hyponormal absolutely norm attaining (or AN-operators in short) and deduce conditions for the normality of the operator. © 2021 Elsevier Masson SAS

Bulletin des Sciences Mathematiques

A representation of hyponormal absolutely norm attaining operators

International Conference cum workshop on Analysis and its Applications June 18–22, 2018 Indian Institute of Technology Madras, Chennai, India

Let H be a right quaternionic Hilbert space and let T be a quaternionic normal operator with domain D(T)⊂H. We prove that there exists a Hilbert basis N of H, a measure space (Ω0,ν), a unitary operator U:H→L2(Ω0;H;ν) and a ν-measurable function η:Ω0→C such that Tx=U⁎MηUx,for allx∈D(T) where Mη is the multiplication operator on L2(Ω0;H;ν) induced by η with U(D(T))⊆D(Mη). We show that every complex Hilbert space can be seen as a slice Hilbert space of some quaternionic Hilbert space and establish the main result by reducing the problem to the complex case then lift it to the quaternion case. © 2020 Elsevier Masson SAS

Spectral theorem for quaternionic normal operators: Multiplication form

Let H be a complex Hilbert space and T: H → H be a bounded linear operator. Then T is said to be norm attaining if there exists a unit vector x0 ϵ H such that ||T x0||= ||T|| f for any closed subspace M of H, the restriction T |M: M → H of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. In this note, we discuss linear maps on B(H), which preserve the class of absolutely norm attaining operators on H. © 2020 Korean Mathematical Society.

Bulletin of the Korean Mathematical Society

Linear maps preserving AN-operators

In this article, we discuss a few spectral properties of paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First, we show that the spectrum of such an operator is non-empty and give a characterization of closed range operators in terms of the spectrum. Using these results, we prove the Weyl’s theorem: if T is a densely defined closed paranormal operator, then σ(T) \ ω(T) = π00(T) , where σ(T),ω(T) and π00(T) denote the spectrum, the Weyl spectrum and the set of all isolated eigenvalues with finite multiplicities, respectively. Finally, we prove that the Riesz projection Eλ with respect to any non-zero isolated spectral value λ of T is self-adjoint and satisfies R(Eλ) = N(T- λI) = N(T- λI) ∗. © 2019, Tusi Mathematical Research Group (TMRG).

Weyl’s theorem for paranormal closed operators

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 give a new characterization of positive absolutely minimum attaining operators in terms of essential spectrum. We discuss absolutely minimum attaining property of multiplication operator on a σ-finite measure space and then establish a spectral decomposition for normal absolutely minimum attaining operators. We also give a sufficient condition under which the adjoint of an absolutely minimum attaining operator is absolutely minimum attaining and show that a paranormal absolutely minimum attaining operator with an extra condition is hyponormal. In the end, we prove a few spectral properties of paranormal operators. © 2019, Tusi Mathematical Research Group (TMRG).

Banach Journal of Mathematical Analysis

Spectral properties of absolutely minimum attaining operators

We give the necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss the structure of such operators in the case of self-adjoint and normal operators separately. Finally, we discuss several properties of absolutely norm attaining operators. © 2019, Indian Academy of Sciences.

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

On absolutely norm attaining operators

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

We study the spectral properties of positive absolutely minimum attaining operators defined on infinite dimensional complex Hilbert spaces and using this we derive a characterization theorem for such type of operators. We construct several examples and discuss some of the properties of this class. Also, we extend this characterization theorem for general absolutely minimum attaining operators by means of the polar decomposition theorem. © 2018 Elsevier Inc.