arXiv subjects

Amit Maji

Publications and source records attributed to Amit Maji.

At least 19 recordsLinked to original sources

arXiv2026

Characterization of paired and Toeplitz + Hankel operators on the polydisc

In this paper, we obtain a complete classification of Toeplitz + Hankel operators on the vector-valued Hardy space $H^2_{\mathcal{E}}(\mathbb{D}^n)$ over the polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$ for $n\geq 1$. We also characterize the paired operators on $L^2(\mathbb{T}^n)$. Furthermore, we give a complete characterization for the class of essentially Toeplitz + essentially Hankel operators on the vector-valued Hardy space $H^2_{\mathcal{E}}(\mathbb{D})$ for finite-dimensional Hilbert space $\mathcal{E}$.

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arXiv2026

Partially isometric truncated and dual truncated Toeplitz operators

Let $\theta$ be a non-constant inner function and let $\phi=\overline{u}v$, where $u$ and $v$ are inner functions such that $v$ divides $\theta$. In this paper we characterize the partially isometric truncated Toeplitz operators $A_{\phi}$ and dual truncated Toeplitz operators $D_{\phi}$ with symbols of the form $\phi=\overline{u}v$. Along with that, we obtain a few more characterization results, including the space of extremal vectors for non-zero partially isometric truncated and dual truncated Toeplitz operators.

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arXiv2025

Invertibility of Bergman Toeplitz operators

In this paper, we establish the invertibility of the Berezin transform of the symbol as a necessary and sufficient condition for the invertibility of the Toeplitz operator on the Bergman space $L^2_a(\mathbb{D})$. More precisely, if ${\phi} = c g + d \bar{g}$, where $c,d\in\mathbb{C}$ and $g\in H^{\infty}(\mathbb{D})$, the space of all bounded analytic functions, then $T_{\phi}$ is invertible on $L^2_a(\mathbb{D})$ if and only if $\inf\limits_{z\in \mathbb{D}}\left|\widetilde{\,{\phi}}(z)\right|=\inf\limits_{z\in \mathbb{D}}|\phi(z)|>0$, where $\widetilde{\,{\phi}}$ is the Berezin transform of $\phi$.

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arXiv2025

Characteristic function of a power partial isometry

The celebrated Sz.-Nagy-Foia\c{s} model theory says that there is a bijection between the class of purely contractive analytic functions and the class of completely non-unitary (c.n.u.) contractions modulo unitary equivalence. In this paper we provide a complete classification of the purely contractive analytic functions such that the associated contraction is a c.n.u. power partial isometry. As an application of our findings, we determine a class of contractive polynomials such that the associated c.n.u. power partial isometry is of the explicit diagonal form $S \oplus N \oplus C$, where $S$ and $C^*$ are unilateral shifts and $N$ is nilpotent. Finally, we obtain a characterization of operator-valued symbols for which the corresponding Toeplitz operator on vector-valued Hardy space is a partial isometry.

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arXiv2025

Power partial isometries

In this paper we obtain a complete characterization of reducing, invariant, and hyperinvariant subspaces for the completely non-unitary component of a power partial isometry. In particular, precise characterization of reducing, invariant, and hyperinvariant subspaces of a truncated shift operator has been achieved.

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arXiv2024

Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$

Let $\mathscr{T}(L^{\infty}(\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^*$-algebra generated by the set $\{T_ϕ : ϕ\in L^{\infty}(\mathbb{T})\}$. Douglas's theorem on symbol map states that there exists a $C^*$-algebra homomorphism from $\mathscr{T}(L^{\infty}(\mathbb{T}))$ onto $L^{\infty}(\mathbb{T})$ such that $T_ϕ\mapsto ϕ$ and the kernel of the homomorphism coincides with commutator ideal in $\mathscr{T}(L^{\infty}(\mathbb{T}))$. In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space $H^2(\mathbb{D}^n)$ over the open unit polydisc $\mathbb{D}^n$ for $n\geq 1$. We further obtain a class of bigger $C^*$-algebras than the Toeplitz algebra $\mathscr{T}(L^{\infty}(\mathbb{T}^n))$ for which the analog of symbol map still holds true.

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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.

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arXiv2024

Orlicz extension of Numerical radius inequalities

In this paper, we achieve new and improved numerical radius inequalities of operators defined on a Hilbert space by using Orlicz function and Hermite-Hadamard inequality. The upper bounds of various inequalities involving numerical radii have been obtained. Finally, we compute an upper bound of the numerical radius for block matrices of the form $\begin{bmatrix}O & P\\Q & O \end{bmatrix}$, where $P, Q$ are any bounded linear operators on a Hilbert space.

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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.

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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.

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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.

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arXiv2021

Doubly commuting mixed invariant subspaces in the polydisc

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is mixed invariant if $M_{z_{j}}(\mathcal{Q}) \subseteq \mathcal{Q}$ for $1 \leq j \leq k$ and $M_{z_{j}}^*(\mathcal{Q}) \subseteq \mathcal{Q}$, $k+1 \leq j \leq n$ for some integer $k \in \{1, 2, \ldots, n-1 \}$. We prove that a mixed invariant subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is doubly commuting if and only if \[ \mathcal{Q} = ΘH^2(\mathbb{D}^k) \otimes \mathcal{Q}_{θ_1} \otimes \cdots \otimes \mathcal{Q}_{θ_{n-k}}, \] where $Θ\in H^{\infty}(\mathbb{D}^k)$ is some inner function and $\mathcal{Q}_{θ_j}$ is either a Jordan block $H^2(\mathbb{D})\ominus θ_j H^2(\mathbb{D})$ for some inner function $θ_j$ or the Hardy space $H^2(\mathbb{D})$. Furthermore, an explicit representation for the commutant of an $n$-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.

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arXiv2018

Pairs of commuting isometries - I

We present an explicit version of Berger, Coburn and Lebow's classification result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of commuting isometric multipliers with precise coefficients. We describe a complete set of (joint) unitary invariants and compare the Berger, Coburn and Lebow's representations with other natural analytic representations of pure pairs of commuting isometries. Finally, we study the defect operators of pairs of commuting isometries.

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arXiv2017

Characterization of Invariant subspaces in the polydisc

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.

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arXiv2017

Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$

We initiate a study of asymptotic Toeplitz operators on the Hardy space $H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$). We also study the Toeplitz operators in the polydisc setting. Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let $T_{z_i}$ denote the multiplication operator on $H^2(\mathbb{D}^n)$ by the $i^{th}$ coordinate function $z_i$, $i =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then the following hold: (i) $T$ is a Toeplitz operator (that is, $T = P_{H^2(\mathbb{D}^n)} M_φ|_{H^2(\mathbb{D}^n)}$, where $M_φ$ is the Laurent operator on $L^{2}(\mathbb{T}^n)$ for some $φ\in L^\infty(\mathbb{T}^n)$) if and only if $T_{z_i}^* T T_{z_i} = T$ for all $i = 1, \ldots, n$. (ii) $T$ is an asymptotic Toeplitz operator if and only if $T = \mbox{~Toeplitz} + \mbox{~compact}$. The case $n = 1$ is the well known results of Brown and Halmos, and Feintuch, respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.

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arXiv2016

Factorizations of Characteristic Functions

Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and $X$ be a row operator from $\oplus_{i=1}^n \mathcal{H}_2$ to $\mathcal{H}_1$. Let $D_{A^*} = (I - A A^*)^{\frac{1}{2}}$ and $D_{B} = (I - B^* B)^{\frac{1}{2}}$ and $Θ_T$ be the characteristic function of $T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B \end{bmatrix}$. Then $Θ_T$ coincides with the product of the characteristic function $Θ_A$ of $A$, the Julia-Halmos matrix corresponding to $L$ and the characteristic function $Θ_B$ of $B$. More precisely, $Θ_T$ coincides with \[ \begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix}, \] where $Γ$ is the full Fock space. Similar results hold for constrained row contractions.

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arXiv2016

Some Paranormed Difference Sequence Spaces of Order $m$ Derived by Generalized Means and Compact Operators

We have introduced a new sequence space $l(r, s, t, p ;Δ^{(m)})$ combining by using generalized means and difference operator of order $m$. We have shown that the space $l(r, s, t, p ;Δ^{(m)})$ is complete under some suitable paranorm and it has Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of this space is computed and also obtained necessary and sufficient conditions for some matrix transformations from $l(r, s, t, p; Δ^{(m)})$ to $l_{\infty}, l_1$. Finally, we obtained some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of some matrix operators on the BK space $l_{p}(r, s, t ;Δ^{(m)})$ by applying the Hausdorff measure of noncompactness.

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arXiv2014

On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals

Let $\boldΦ=(ϕ_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold Φ}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\boldΦ}^{A}(X)$ is a $σ$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\boldΦ}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.

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