arXiv subjects

Satyabrata Majee

Publications and source records attributed to Satyabrata Majee.

5 recordsLinked to original sources

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.

math.FA↗

arXiv2024

Wold decomposition for isometries with equal range

Let $n \geq 2$, and let $V=(V_1,\dots, V_n)$ be an $n$-tuple of isometries acting on a Hilbert space $\mathcal{H}$. We say that $V$ is an $n$-tuple of isometries with equal range if $V_i^{m_i}V_j^{m_j}\mathcal{H} = V_j^{m_j} V_i^{m_i}\mathcal{H}$ and $V_i^{*m_i}V_j^{m_j} \mathcal{H} = V_j^{m_j} V_i^{*m_i}\mathcal{H}$ for $m_i,m_j \in \mathbb{Z}_+$, where $1 \leq i<j \leq n$. We prove that each $n$-tuple of isometries with equal range admits a unique Wold decomposition. We obtain analytic models of the above class, and as a consequence, we show that the wandering data are complete unitary invariants for $n$-tuples of isometries with equal range. Our results unify all prior findings on the decomposition for tuples of isometries in the existing literature.

math.FA↗

arXiv2023

Wold-type decomposition for $\mathcal{U}_n$-twisted contractions

Let $n>1$, and $\{U_{ij}\}$ for $1 \leq i < j \leq n$ be $\binom{n}{2}$ commuting unitaries on a Hilbert space $\mathcal{H}$ such that $U_{ji}:=U^*_{ij}$. An $n$-tuple of contractions $(T_1, \dots, T_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted contraction with respect to a twist $\{U_{ij}\}_{i<j}$ if $T_1, \dots, T_n$ satisfy \[ T_iT_j=U_{ij}T_jT_i; \hspace{0.5cm} \hspace{1cm} T_i^*T_j= U^*_{ij}T_jT_i^* \hspace{0.5cm} \mbox{and} \hspace{0.5cm} T_kU_{ij} =U_{ij}T_k \] for all $i,j,k=1, \dots, n$ and $i \neq j$. We obtain a recipe to calculate the orthogonal spaces of the Wold-type decomposition for $\mathcal{U}_n$-twisted contractions on Hilbert spaces. As a by-product, a new proof as well as complete structure for $\mathcal{U}_2$-twisted (or pair of doubly twisted) and $\mathcal{U}_n$-twisted isometries have been established.

math.FA↗

arXiv2022

On decomposition for pairs of twisted contractions

This paper presents Wold-type decomposition for various pairs of twisted contractions on Hilbert spaces. As a consequence, we obtain Wold-type decomposition for pairs of doubly twisted isometries and in particular, new and simple proof of Słoćinski's theorem for pairs of doubly commuting isometries are provided. We also achieve an explicit decomposition for pairs of twisted contractions such that the c.n.u. parts of the contractions are in $C_{00}$. It is shown that for a pair $(T,V^*)$ of twisted operators with $T$ as a contraction and $V$ as an isometry, there exists a unique (upto unitary equivalence) pair of doubly twisted isometries on the minimal isometric dilation space of $T$. As an application, we prove that pairs of twisted operators consisting of an isometry and a co-isometry are doubly twisted. Finally, we have given a characterization for pairs of doubly twisted isometries.

math.FA↗

arXiv2022

Numerical radius and Berezin number inequality

We study various inequalities for numerical radius and Berezin number of a bounded linear operator on a Hilbert space. It is proved that the numerical radius of a pure two-isometry is 1 and the Crawford number of a pure two-isometry is 0. In particular, we show that for any scalar-valuednon-constant inner function $θ$, the numerical radius and the Crawford number of a Toeplitz operator $T_θ$ on a Hardy space is 1 and 0, respectively. It is also shown that numerical radius is multiplicative for a class of isometries and sub-multiplicative for a class of commutants of a shift. We have illustrated these results with some concrete examples. Finally, some Hardy-type inequalities for Berezin number of certain class of operators are established with the help of the classical Hardy's inequality.

math.FA↗