arXiv subjects

Puspendu Nag

Publications and source records attributed to Puspendu Nag.

4 recordsLinked to original sources

arXiv2026

Norm and Minimum Attainment of Foguel-Type Operators

In this article, we study Foguel-type operators $\EuScript{F}=\begin{bmatrix} V_1^{*} & T\\ 0 & V_2\end{bmatrix}$ on $H_1\oplus H_2$, where each $H_j$ is a separable Hilbert space, $V_j\in \mathcal{B}(H_j)$ is an isometry for $j=1,2$, and $T\in \mathcal{B}(H_2,H_1)$. We examine how the norm and minimum attainment properties of $\EuScript{F}$ depend on the corner entry $T$ and on the defect spaces of the isometries. The norm formula $\|\EuScript{F}\|=\frac{\|T\|+\sqrt{\|T\|^{2}+4}}{2}$ holds for every $T$ when $V_1,V_2$ are isometries, and more generally when $V_2$ is a contraction that is isometric on the range of $T^{*}$. Moreover, $\EuScript{F}$ attains its norm if and only if $T$ does. We characterize the positive $2\times 2$ block operator matrices that are absolutely norm attaining and absolutely minimum attaining, and use this to prove that $\EuScript{F}$ is absolutely norm attaining if and only if it is absolutely minimum attaining if and only if $\dim N(V_1^{*})<\infty$ and $T$ has finite rank. The closure of the absolutely norm attaining class is described by $\dim N(V_1^{*})<\infty$ and compactness of $T$. Further, we establish that the reduced minimum modulus is $γ(\EuScript{F})=\frac{\sqrt{\|T\|^{2}+4}-\|T\|}{2}=\frac{1}{\|\EuScript{F}\|}$, and $\EuScript{F}$ attains it if and only if $T$ is norm attaining. We also show that $\EuScript{F}$ is Fredholm if and only if $\dim N(V_j^{*})<\infty$ for $j=1,2$, with index $\dim N(V_1^{*})-\dim N(V_2^{*})$, and we give its Moore--Penrose inverse. Finally, for $V_1=V_2=T_z$ and more generally $V_j=T_{θ_j}$ with $θ_j$ inner, and Toeplitz or Hankel corner entries, the results become explicit conditions on the symbol.

math.FA↗

arXiv2026

Norm attaining dual truncated Toeplitz operators

In this paper, we investigate norm attainment for dual truncated Toeplitz operators $D_\vp$ acting on $\clk_u^\perp=uH^2\oplus H^2_{-}$, where $u$ is a nonconstant inner function and $\vp\in L^\infty(\T)$. Our main focus is the structure of extremal vectors and the distinction between global and componentwise norm attainment. For arbitrary $\vp\in L^\infty(\T)$, we establish an exact norm-defect identity and characterize the extremal space in terms of the essential maximum set $E_\vp=\{ζ\in\T:|\vp(ζ)|=\|\vp\|_\infty\}$. As a consequence, when $u$ is a finite Blaschke product, \[ D_\vp\in\mathcal{NA}\quad\Longleftrightarrow\quad m(E_\vp)>0. \] In this case, whenever $D_\vp$ is norm attaining, its extremal space is infinite-dimensional. We further show that the sets of symbols generating norm attaining and non-norm attaining DTTOs are both norm dense in $L^\infty(\T)$. Consequently, both the norm attaining and the non-norm attaining DTTOs are operator-norm dense in the class of all DTTOs associated with $u$. For unimodular symbols, we characterize extremality by the condition $M_\vp f\in\clk_u^\perp$ and equivalently by a truncated Hankel kernel condition. For mixed extremal vectors $f=x\oplus y$, we derive the identity \[ \|D_\vp x\|^2-\|x\|^2=\|D_\vp y\|^2-\|y\|^2=-\langle D_\vp x,D_\vp y\rangle, \] which yields a phase-rotation criterion and coupled Toeplitz--Hankel relations. We also show that global norm attainment may occur even when neither the analytic nor the coanalytic component contains a nonzero extremal vector. Under additional Hardy-space hypotheses, we obtain factorization criteria for componentwise extremals, construct explicit extremal families for quotient-inner symbols, and relate norm attainment of Toeplitz operators to that of dual truncated Toeplitz operators.

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) $D_φ$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $φ\in L^\infty(\T)$. 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 $m(D_φ)$ coincides with the essential infimum of $|φ|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_φ)$ 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↗

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↗