Anderson's theorem and A-spectral radius bounds for semi-Hilbertian space operators

Bhunia, Pintu; Kittaneh, Fuad; Paul, Kallol; Sen, Anirban

doi:10.1016/j.laa.2022.10.019

Cited by 13 publications

(9 citation statements)

References 22 publications

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“…Recently, Anderson's theorem and its compact extension has been studied in the setting of semi-Hilbertian operators, see [9]. Our main goal of this section is to generalize [11,Th.…”

Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 99%

“…For more details about B A (H) and B A 1/2 (H) we refer readers to [1,2,3,18]. The notion of A-compact operators generalizing the compact operators was introduced and studied in [9]. Let K A 1/2 (H) denote the collection of all A-compact operators.…”

Section: Introductionmentioning

confidence: 99%

“…2.1]). The A-numerical radius and A-spectral radius inequalities have been studied by many mathematicians [9,10,17,25,28,33] over the years. To the best of our knowledge, there is hardly any literature available on the A-numerical range of semi-Hilbertian operators.…”

Section: Introductionmentioning

confidence: 99%

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Sen¹,

Goswami²

2023

Redefining Virtual Teaching Learning Pedagogy

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In this paper we explore the relation between the A-numerical range and the A-spectrum of A-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of A-normal operator and prove that closure of the A-numerical range of an A-normal operator is the convex hull of the A-spectrum. We further prove Anderson's theorem for the sum of A-normal and A-compact operators which improves and generalizes the existing result on Anderson's theorem for A-compact operators. Finally we introduce strongly A-numerically closed class of operators and along with other results prove that the class of A-normal operators is strongly A-numerically closed.

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“…Recently, Anderson's theorem and its compact extension has been studied in the setting of semi-Hilbertian operators, see [9]. Our main goal of this section is to generalize [11,Th.…”

Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Sen¹,

Goswami²

2023

Redefining Virtual Teaching Learning Pedagogy

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“…Further, the range and the kernel of T are denoted by R(T ) and N(T ), respectively. In addition, the cone of all positive operators on H is given by on H if and only if A is injective, and that (H, • A ) is complete if and only if R(A) is a closed subspace of H. For very recent contributions concerning operators acting on semi-Hilbert spaces, we refer the reader to [2,6,9,11] and the references therein. From now on, we suppose that A ∈ B(H) is always a positive (nonzero) operator and we denote the A-unit sphere of H by S A (0, 1), that is, S A (0, 1) := {x ∈ H ; x A = 1}.…”

Section: Introductionmentioning

confidence: 99%

Inequalities for the $A$-joint numerical radius of two operators and their applications

Feki

2024

Hacettepe Journal of Mathematics and Statistics

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Let $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ and defines a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} \omega_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx, x\rangle}_A\big|^2+\big|{\langle Sx, x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving $\omega_{A,\text{e}}(T,S)$. This allows us to establish some inequalities for the $A$-numerical radius of $A$-bounded operators. In particular, we extend the well-known inequalities due to Kittaneh [Studia Math. 168 (2005), no. 1, 73-80]. Moreover, several bounds related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators are also provided.

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“…SIP had been applied for the solution of general quadratic programming [14], also as a representation of quiver [7]. Latest studies on SIP showed the generalization of concepts in IP space to SIP space, such as properties of normal operators ( [11], [5]), isometry and unitary properties [1], and closed operator [4]; also, geometrical aspects, such as metric on projections [2], Birkhoff-James orthogonality ( [17], [12]) and numerical radius ( [8], [6], [13]).…”

Section: Introductionmentioning

confidence: 99%

On The Adjoint of Bounded Operators On A Semi-Inner Product Space

Respitawulan,

Pangestu,

Yuliawan

et al. 2023

JIMS

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The notion of semi-inner product (SIP) spaces is a generalization of inner product (IP) spaces notion by reducing the positive definite property of the product to positive semi-definite. As in IP spaces, the existence of an adjoint of a linear operator on a SIP space is guaranteed when the operator is bounded. However, in contrast, a bounded linear operator on SIP space can have more than one adjoint linear operators. In this article we give an alternative proof of those results using the generalized Riesz Representation Theorem in SIP space. Further, the description of all adjoint operators of a bounded linear operator in SIP space is identified.

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Anderson's theorem and A-spectral radius bounds for semi-Hilbertian space operators

Cited by 13 publications

References 22 publications

Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Inequalities for the $A$-joint numerical radius of two operators and their applications

On The Adjoint of Bounded Operators On A Semi-Inner Product Space

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