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Arup Majumdar

Publications and source records attributed to Arup Majumdar.

9 recordsLinked to original sources
arXiv2026

Unbounded Antilinear Operators on Hilbert Spaces

The paper introduces unbounded antilinear operators on Hilbert spaces and develops their fundamental theory. In particular, we establish a closed range theorem, a polar decomposition theorem, and the convexity of the numerical range for antilinear operators. Furthermore, we present several new results on antilinear normal operators and provide necessary and sufficient conditions for the existence of a minimal antilinear normal extension of an antilinear subnormal operator. We further develop a comprehensive characterization of antilinear block operator matrices with purely antilinear entries, establishing necessary and sufficient criteria for their closability through the framework of Schur and quadratic complements.

math.FA↗
arXiv2026

Spectral Properties of Off Diagonal Block Linear Relations via Moore Penrose Inverses in Hilbert Spaces

In this paper, we characterize the essential spectra and the resolvent set of the off-diagonal block linear relation \[ \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \] in terms of the essential spectra and resolvent sets of the products $AB$ and $BA$. Our approach establishes precise spectral relationships that connect the structural properties of the block linear relation with those of the associated compositions. Furthermore, we investigate the Moore--Penrose inverses of closed linear relations in Hilbert spaces and employ these results to extend the spectral analysis to the off-diagonal block linear relation \[ \mathcal{A} = \begin{bmatrix} 0 & T^{\dagger} \\ T & 0 \end{bmatrix}, \] where $T$ is a closed, continuous linear relation with closed range from a Hilbert space $H$ to a Hilbert space $K$, and $T^{\dagger}$ denotes its Moore--Penrose inverse.

math.FA↗
arXiv2025

Hyers-Ulam stability of closed linear relations in Hilbert spaces

This paper introduces the concept of Hyers-Ulam stability for linear relations in normed linear spaces and presents several intriguing results that characterize the Hyers-Ulam stability of closed linear relations in Hilbert spaces. Additionally, sufficient conditions are established under which the sum and product of two Hyers-Ulam stable linear relations remain stable.

math.FA↗
arXiv2024

On the generalized Cauchy dual of closed operators in Hilbert spaces

In this paper, we introduce the generalized Cauchy dual $w(T) = T(T^{*}T)^{\dagger}$ of a closed operator $T$ with the closed range between Hilbert spaces and present intriguing findings that characterize the Cauchy dual of $T$. Additionally, we establish the result $w(T^{n}) = (w(T))^{n}$, for all $n \in \mathbb{N}$, where $T$ is a quasinormal EP operator.

math.FA↗
arXiv2024

Characterizations of closed EP operators on Hilbert spaces

In this paper, we present intriguing findings that characterize both the closed (unbounded) and bounded EP operators on Hilbert spaces. Additionally, we demonstrate the result $γ(T) \leq r(T)$, where $T$ is a bounded EP operator, and $γ(T) \text{ and } r(T)$ represent the reduced minimum modulus and the spectral radius of $T$, respectively.

math.FA↗
arXiv2024

Hyers-Ulam Stability of Unbounded Closable Operators in Hilbert Spaces

In this paper, we discuss the Hyers-Ulam stability of closable (unbounded) operators with several interesting examples. We also present results pertaining to the Hyers-Ulam stability of the sum and product of closable operators to have the Hyers-Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of $2 \times 2$ block matrix $\mathcal A$ in order to have the Hyers-Ulam stability.

math.FA↗
arXiv2024

$A$-approximate point spectrum of $A$-bounded operators in semi-Hilbertian spaces

This paper delves into several characterizations of $A$-approximate point spectrum of A-bounded operators acting on a complex semi-Hilbertian space $H$ and also investigates properties of the $A$-approximate point spectrum for the tensor product of two $A^{\frac{1}{2}}$-adjoint operators. Furthermore, several properties of $A$-normal operators have been established.

math.FA↗
arXiv2024

A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces

In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |λ|: λ\in σ_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |λ|: λ\in σ_{A}(T)\} = r_{A}(T), \text{ when } T \in B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte spectrum $σ_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}= (T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that $r_{A_{h}}(\mathbf{T}) \leq \sup \{\|λ\|_{2}: λ\in σ_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte spectral radius of $\mathbf{T}$.

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