arXiv subjects

Kousik Dhara

Publications and source records attributed to Kousik Dhara.

7 recordsLinked to original sources

arXiv2025

Toeplitz, Hankel, de Branges and two truncated matrix moment problems

This paper deals with (1) the truncated matrix Hamburger moment problem from the point of view of reproducing kernel Hilbert spaces of vector valued entire functions of the kind introduced and extensively studied by Louis de Branges and (2) the truncated matrix trigonometric moment problem viewed through an analogous class of spaces that are formulated with respect to the open unit disc rather than the open upper half-plane. In this approach projections are computed via appropriately chosen reproducing kernels instead of orthogonal bases. This approach eases the bookkeeping and leads to pleasing formulas.

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arXiv2024

A Bishop-Phelps-Bollobás theorem for bounded analytic functions

Let $H^\infty$ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by $\mathscr{B}(H^\infty)$ the Banach space of all bounded linear operators from $H^\infty$ to itself. We prove that the Bishop-Phelps-Bollobás property holds for $\mathscr{B}(H^\infty)$. As an application to our approach, we prove that the Bishop-Phelps-Bollobás property also holds for operator ideals of $\mathscr{B}(H^\infty)$.

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arXiv2024

The Bishop-Phelps-Bollobas property for certain Banach spaces

Let $X$ be a complex Banach space. We prove that if $L$ is an extremally disconnected compact Hausdorff topological space, then the pair $(X, C(L))$ satisfies the Bishop-Phelps-Bollobás property (BPBp for short). As a byproduct, we obtain the BPBp for the pair $(X, L^\infty(ν))$ for any measure $ν$. In particular, this settles an unresolved question regarding the BPBp for the pair $(L^\infty(μ), L^\infty(ν) )$ for any two measures $μ$ and $ν$. Finally, we show that $(X,H^\infty(Ω)$ has the BPBp when $Ω$ is a multi-connected planar domain bounded by finitely many disjoint analytic simple closed curves.

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arXiv2022

Norms of basic operators in vector valued model spaces and de Branges spaces

Let $Ω_+$ be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator $A_α=Π_ΘT_{b_α}|_{\mathcal{H}(Θ)}$ in the vector valued model space $\mathcal{H}(Θ)=H^m_2 \ominus ΘH^m_2$ associated with an $m\times m$ matrix valued inner function $Θ$ in $Ω_+$ and show that the norm is attained. Here $Π_Θ$ denotes the orthogonal projection from the Lebesgue space $L^m_2$ onto $\mathcal{H}(Θ)$ and $T_{b_α}$ is the operator of multiplication by the elementary Blaschke factor $b_α$ of degree one with a zero at a point $α\in Ω_+$. We show that if $A_α$ is strictly contractive, then its norm may be expressed in terms of the singular values of $Θ(α)$. We then extend this evaluation to the more general setting of vector valued de Branges spaces.

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arXiv2022

Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} Λ_{\varepsilon}^{S}(T) := σ_S (T) \bigcup \left \{ q \in \mathbb{H}\setminus σ_S(T):\; \|Δ_{q}(T)^{-1}\| \geq \frac{1}{\varepsilon} \right\}, \end{align*} where $\mathbb{H}$ denotes the division ring of quaternions, $σ_S(T)$ is the $S$-spectrum of $T$ and $Δ_q(T)= T^2-2 \text{Re}(q)T+|q|^2I$. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo $S$-spectrum and explicitly compute the pseudo $S$-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of $S$-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number $r$ if and only if for every $\varepsilon>0$ the pseudo $S$-spectrum of the operator is the circularization of a closed disc in the complex plane centered at $r$ with the radius $\sqrt{\varepsilon}$. Further, we propose a $G_1$-condition for quaternionic operators and prove some results in this setting.

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arXiv2021

Idempotent, model, and Toeplitz operators attaining their norms

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$ attains its norm. Here $S_{\mathcal{Q}} = P_{\mathcal{Q}}M_z|_{\mathcal{Q}}$, the compression of the shift $M_z$ on the Hardy space $H^2(\mathbb{D})$ to $\mathcal{Q}$.

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arXiv2020

Orthogonality in Banach Spaces via projective tensor product

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+λy\|_{X} \geq \|x\|_{X}$ for all $λ\in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $φ$ on $X$ such that $\|φ\| = 1$, $φ(x,x)=\|x\|^2$ and $φ(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.

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