arXiv subjects

Sudip Ranjan Bhuia

Publications and source records attributed to Sudip Ranjan Bhuia.

8 recordsLinked to original sources

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.

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arXiv2026

Nearly invariant subspaces and weighted dual truncated Toeplitz operators

Let $\mathcal{M}=h\mathcal{K}_u$ be a nearly $S^*$-invariant subspace of $H^2$, where $\mathcal{K}_u=H^2\ominus uH^2$ and $h$ is the extremal multiplier. For $\vp\in L^\infty(\T)$, we study the compression \[ D_\vp^\mathcal{M} = \restr{P_{\mathcal{M}^\perp}M_\vp}{\mathcal{M}^\perp}, \] called a weighted dual truncated Toeplitz operator. When $h\equiv1$, this reduces to the classical dual truncated Toeplitz operator. Using the Hartmann--Ross projection formula, we prove \[ \|D_\vp^\mathcal{M}\|=\|\vp\|_\infty, \] characterize compactness, and show that the natural multiplication map by $h$ identifies the weighted and classical theories precisely when $h$ is inner. We also establish complex symmetry and obtain block matrix, defect, and semi-commutator identities via weighted truncated Hankel operators. As a main algebraic consequence, we prove \[ D_\vp^\mathcal{M} D_ψ^\mathcal{M}=0 \quad\Longleftrightarrow\quad \vp=0\ \text{or}\ ψ=0 \quad\text{a.e. on }\T. \] Finally, we derive a a rank-at-most-two correction formula and a finite-rank displacement identity for the generalized dual shift $D_z^\mathcal{M}$.

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arXiv2026

Algebraic characterizations of paired operators on model spaces

We study paired multiplication operators associated with orthogonal decompositions of $L^2(\mathbb{T})$ arising from Hardy spaces and Sz.-Nagy--Foias model spaces. For a non-constant inner function $θ$, we characterize $\mathcal{K}_θ$-paired and triply paired operators by finite-rank commutator identities with the bilateral shift $M_z$. We also apply these ideas to sums of truncated Toeplitz and truncated Hankel operators on $\mathcal{K}_θ$. Using the displacement characterizations of Sarason and Gu--Ma, we derive a second-order finite-rank displacement formula for such sums and describe the ambiguity of the decomposition in terms of $\mathcal{I}_θ=\mathcal{T}_θ\cap \mathcal{H}_θ$. In the finite-dimensional case $θ(z)=z^n$, the framework recovers the classical Toeplitz-plus-Hankel matrix recurrence of Bevilacqua--Bonanni--Bozzo.

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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.

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arXiv2024

Operators commuting with complex symmetric weighted composition operators on $H^2$

In this paper, we initially study when an anti-linear Toeplitz operator is in the commutant of a composition operator. Primarily, we investigate weighted composition operators $W_{g,ψ}$ commuting with complex symmetric weighted composition operators $W_{f,φ}$ on the Hardy space $H^2(\mathbb{D})$. In particular, we give the descriptions of the symbols $g$ and $ψ$ such that the inducing weighted composition operator $W_{g,ψ}$ commutes with the complex symmetric weighted composition operator $W_{f,φ}$ with the conjugation $\mathcal{J}$. Furthermore, we subsequently demonstrate that these weighted composition operators are normal and complex symmetric in accordance with the properties of the fixed point of the associated symbol $φ$.

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arXiv2024

Factorization of anti-linear and $C$-normal operators

A conjugation $C$ is an anti-linear isometric involution on a complex Hilbert space $\clh$, and $T\in \clb(\clh)$ is conjugate normal if $T^*T = CTT^*C$ holds for some conjugation (C). In this paper, we provide a factorization and range inclusion theorem for anti-linear operators, and consequently, establish the polar decomposition for anti-linear operators by applying the Douglas theorem on majorization of Hilbert space operators. Moreover, we present a factorization of $C$-normal operators based on the polar decomposition. Lastly, we study the Cartesian decomposition of conjugate normal operators, thereby expanding the results in [18].

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arXiv2024

Norm attaining composition operators on Segal-Bargmann spaces

In this note, we study the composition operators on Segal-Bargmann spaces, which attains its norm and we show that every composition operators on the classical Fock space over $\mathbb{ C}^n$ is norm attaining. Also, we establish a necessary and sufficient condition for a sum of two kernel functions to be an extremal function for the norm of composition operators.

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arXiv2022

Characterizations of complex symmetric Toeplitz operators

We present complete characterizations of Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is $S$-Toeplitz for some unilateral shift $S$ and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift $S$. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results answer the well known open question about characterizations of complex symmetric Toeplitz operators.

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