arXiv subjects

Puspendu Nag

Publications and source records attributed to Puspendu Nag.

4 recordsLinked to original sources

arXiv2026

Minimum attaining operators on reducing subspaces: Spectral structure and density

In this article, we introduce and investigate a new subclass $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$. This class contains the absolutely minimum attaining operators and is properly contained in the class of minimum attaining operators. We establish several structural and spectral characterizations of operators in $\mathcal{M}_r(H)$. In particular, we characterize positive operators in $\mathcal{M}_r(H)$ in terms of their spectral representations. We prove that $\mathcal{M}_r(H)$ is dense in $\mathcal{B}(H)$ in the operator norm and, moreover, that the operators in $\mathcal{M}_r(H)$ having a nontrivial invariant half-space are also dense in $\mathcal{B}(H)$ in the operator norm. We further obtain a representation theorem for normal operators in $\mathcal{M}_r(H)$ and establish additional structural properties of this class.

math.FA↗

arXiv2026

On the minimum modulus of dual truncated Toeplitz operators

This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_{\varphi}$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $\varphi \in L^\infty(\T)$. We first establish an explicit formula for the minimum modulus of the compressed shift $S_u$ and its dual $D_u$ in terms of $|u(0)|$, and prove that the minimum is always attained. For normal DTTOs, we derive sharp spectral bounds utilizing the essential range of the symbol and characterize the conditions under which $m(D_{\varphi})$ coincides with the essential infimum of $|\varphi|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_{\varphi})$ by analyzing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results.

math.FA↗

arXiv2026

Norm attaining dual truncated Toeplitz operators

This paper develops a complete framework for understanding when a dual truncated Toeplitz operator (DTTO) attains its norm. Given a nonconstant inner function $u$, the DTTO associated with a symbol $\varphi \in L^{\infty}(\mathbb{T})$ acts on the orthogonal complement ${\mathcal{K}_u}^{\perp} = uH^{2} \oplus H^{2}_{-}$ of the model space $\mathcal{K}_u = H^{2}\ominus uH^{2}$. Assuming $\|\varphi\|_{\infty}=1$, we give a characterization of the norm attaining property of $D_{\varphi}$ and describe all extremal vectors. A sharp analytic and coanalytic dichotomy emerges $D_{\varphi}$ attains its norm precisely when the symbol admits either $\varphi=\overline{u}\overline{\psi}_{+}\chi_{+}$ or $\varphi=u\psi_{-}\overline{\chi}_{-},$ where $\psi_{\pm},\chi_{\pm}$ are inner functions. The first condition corresponds to norm attainment on the analytic component $uH^{2}$, while the second corresponds to norm attainment on the coanalytic component $H^{2}_{-}$ via the natural conjugation $C_{u}$. A key feature of the theory is that the dual compressed shift $D_{u}$ (the case $\varphi(z)=z$) always attains its norm. We also obtain a coupled Toeplitz, Hankel system governing analytic and coanalytic components of extremal vectors, and provide several concrete examples including nonanalytic unimodular symbols illustrating how the factorization criteria govern norm attainment.

math.FA↗

arXiv2025

On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix

In this article, we study absolutely norm attaining operators ($\mathcal{AN}$-operators, in short), that is, operators that attain their norm on every non-zero closed subspace of a Hilbert space. Our focus is primarily on positive $2\times2$ block operator matrices in Hilbert spaces. Subsequently, we examine the analogous problem for operators that attain their minimum modulus on every nonzero closed subspace; these are referred to as absolutely minimum attaining operators (or $\mathcal{AM}$-operators, in short). We provide conditions under which these operators belong to the operator norm closure of the above two classes. In addition, we give a characterization of idempotent operators that fall into these three classes. Finally, we illustrate our results through examples that involve concrete operators.

math.FA↗