arXiv subjects

Bappa Bisai

Publications and source records attributed to Bappa Bisai.

6 recordsLinked to original sources

arXiv2022

A generalization of Ando's dilation, and isometric dilations for a class of tuples of $q$-commuting contractions

Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or }T_1T_2=T_2T_1Q. \] We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=qI_{\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of isometric dilations for pairs of $q$-commuting contractions which are well studied. There is an extended notion of $q$-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an $n$-tuple of $q$-commuting contractions, where $n\geq 3$. Generalizing the class of commuting contractions considered by Brehmer [8], we construct a class of $n$-tuples of $q$-commuting contractions and find isometric dilations explicitly for the class.

math.FA↗

arXiv2022

Admissible fundamental operators associated with two domains related to $μ$-synthesis

In this article, we discuss necessary condition of conditional dilation for both completely non-unitary (c.n.u) $Γ_{n}$-contractions and c.n.u $\mathbb E$-contractions. Consider two tuples, $(A_1, \dots, A_{n-1})$ and $(B_1, \dots, B_{n-1})$, of operators defined on two Hilbert spaces. One of the principal goals is to identify a necessary and a sufficient condition guaranteeing the existence of a c.n.u $Γ_n$-contraction $(S_1, \dots, S_{n-1},P)$ such that $(A_1, \dots, A_{n-1})$ becomes the $\mathcal{F}_O$-tuple of $(S_1, \dots, S_{n-1},P)$ and $(B_1, \dots,B_{n-1})$ becomes the $\mathcal{F}_O$-tuple of $(S_1^*,\dots, S_{n-1}^*,P^*)$. Also for given two pairs of operators $(F_1,F_2)$ and $(G_1,G_2)$ defined on two Hilbert spaces, we examine when there is a c.n.u $\mathbb E$-contraction $(A,B,P)$ such that $(F_1,F_2)$ becomes the $\mathcal{F}_O$-pair of $(A,B,P)$ and $(G_1,G_2)$ becomes the $\mathcal{F}_O$-pair of $(A^*,B^*,P^*)$.

math.FA↗

arXiv2022

A Nagy-Foias program for a c.n.u. $Γ_n$-contraction

A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) : |z_i|\leq 1\,, \; \; \; 1\leq i \leq n-1 \right\} \] as a spectral set is called a $Γ_n$-contraction. From the literature we have that a point $(s_1, \dots , s_{n-1},p)$ in $Γ_n$ can be represented as $s_i=c_i+pc_{n-i}$ for some $(c_1, \dots, c_{n-1}) \in Γ_{n-1}$. We construct a minimal $Γ_n$-isometric dilation for a particular class of c.n.u. $Γ_n$-contractions $(S_1, \cdots, S_{n-1},P)$ and obtain a functional model for them. With the help of this model we express each $S_i$ as $S_i=C_i+PC_{n-i}$, which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. $Γ_n$-contractions satisfying $S_i^*P=PS_i^*$ for each $i$. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that $S_i^*P=PS_i^*$. We apply this abstract model to achieve a complete unitary invariant for such c.n.u. $Γ_n$-contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple $(S_1, \dots , S_{n-1},P)$ becomes a $Γ_n$-contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction.

math.FA↗

arXiv2022

On $q$-commuting co-extensions and $q$-commutant lifting

Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying $TX=qXT$ for a complex number $q$. There are some interesting results in the literature on $q$-commuting dilation and $q$-commutant lifting of such pair $(T,X)$ when $|q|=1$. Here we improve a few of them to the class of scalars $q$ satisfying $|q|\leq \dfrac{1}{\|T\|}$.

math.FA↗

arXiv2021

Automorphisms and the fundamental operators associated with the symmetrized tridisc

The automorphisms of the symmetrized polydisc $\mathbb G_n$ are well-known and are given in the coordinates of the polydisc in \cite{E:Z}. We find an explicit formula for the automorphisms of $\mathbb G_n$ in its own coordinates. If $τ$ is an automorphism of $\mathbb G_n$, then $τ(S_1,\dots,S_{n-1},P)$ is a $Γ_n$-contraction, where a $Γ_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $Γ_n$ is a spectral set. Corresponding to every $Γ_n$-contraction $(S_1,\dots,S_{n-1},P)$, there exist $n-1$ unique operators $A_1,\dots,A_{n-1}$ such that \[ S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad D_P=(I-P^*P)^{1/2}\,, \] for $i=1,\dots, n-1$. This unique $(n-1)$-tuple $(A_1,\dots,A_{n-1})$, which is called the fundamental operator tuple or $\mathcal F_O$-tuple of $(S_1,\dots,S_{n-1},P)$ in literature, plays central role in every section of operator theory on $Γ_n$. We find an explicit form of the $\mathcal F_O$-tuple of $τ(S_1,\dots,S_{n-1},P)$ when $n=3$. We show by an example that a $Γ_n$-contraction may not have commuting $\mathcal F_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $Γ_n$-contractions are unitarily equivalent.

math.FA↗

arXiv2019

Structure theorems for operators associated with two domains related to $μ$-synthesis

A commuting tuple of $n$ operators $(S_1, \dots, S_{n-1}, P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots, \prod_{i=1}^{n}z_i \right) : |z_i|\leq 1, i=1, \dots, n \right\} \] is a spectral set is called a $Γ_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called an $\mathbb E$-contraction, where \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3 \neq 0 \quad \forall z, w \in \overline{\mathbb D} \}. \] There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both $Γ_n$-contractions and $\mathbb E$-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component $P$. The central role in determining the structure of a $Γ_n$-contraction or an $\mathbb E$-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a $Γ_n$-contraction or an $\mathbb E$-contraction.

math.FA↗