arXiv subjects

Haripada Sau

Publications and source records attributed to Haripada Sau.

26 records · Page 2Linked to original sources

arXiv2019

Toeplitz operators and pseudo-extensions

There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by studying the unique canonical unitary pseudo-extension of a tuple of commuting contractions. The third one is about the $C^*$-algebra generated by the Toeplitz operators determined by a commuting tuple of contractions. With the help of a special completely positive map, a different proof of the existence of the unique canonical unitary pseudo-extension is given.

math.FA↗

arXiv2018

Functional Models and Invariant Subspaces for Pairs of Commuting Contractions

The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equivalence. In addition to the characteristic function $Θ_T$, we identify additional invariants $({\mathbb G}, {\mathbb W})$ which can be used to construct a functional model for the commuting pair $(T_1, T_2)$ and which have good uniqueness properties: if two commutative contractive pairs $(T_1, T_2)$ and $(T'_1, T'_2)$ are unitarily equivalent, then their characteristic triples $(Θ, {\mathbb G}, {\mathbb W})_T$ and $(Θ, {\mathbb G}, {\mathbb W})_{T'}$ coincide in a natural sense. We illustrate the theory with several simple cases where the characteristic triples can be explicitly computed. This work extends earlier results of Berger-Coburn-Lebow \cite{B-C-L} for the case where $(T_1, T_2)$ is a pair of commuting isometries, and of Das-Sarkar \cite{D-S}, Das-Sarkar-Sarkar \cite{D-S-S} and the second author \cite{sauAndo} for the case where $T = T_1T_2$ is pure (the operator sequence $T^{*n}$ tends strongly to $0$). Finally we use the model to study the structure of joint invariant subspaces for a commutative, contractive operator pair, extending results of Sz.-Nagy--Foias for the single-operator case.

math.FA↗

arXiv2018

Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

One of the most important results in operator theory is Andô's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schäffer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Andô dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of Andô's dilation theorem by giving both Schäffer-type and Douglas-type explicit constructions of an Andô dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction $T$ into the product of two commuting contractions. Unlike the one-variable case, two minimal Andô dilations need not be unitarily equivalent. However, we show that the compressions of the two Andô dilations constructed in this paper to the minimal dilation spaces of the contraction $T_1T_2$, are unitarily equivalent. In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to 0$ strongly, an Andô dilation was constructed recently in \cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure contraction.

math.FA↗

arXiv2016

Holomorphic functions on the symmetrized bidisk - realization, interpolation and extension

There are three new things in this paper about the open symmetrized bidisk $\mathbb G = \{(z_1+z_2, z_1z_2) : |z_1|, |z_2| < 1\}$. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extension Theorem: A characterization is obtained of those subsets $V$ of the open symmetrized bidisk $\mathbb G$ that have the property that every function $f$ holomorphic in a neighbourhood of $V$ and bounded on $V$ has an $H^\infty$-norm preserving extension to the whole of $\mathbb G$. \end{enumerate}

math.FA↗

arXiv2016

$Γ$-unitaries, dilation and a natural example

This note constructs an explicit normal boundary dilation for a commuting pair $(S,P)$ of bounded operators with the symmetrized bidisk $$Γ=\{(z_1+z_2,z_1z_2):|z_1|,|z_2| \leq 1\}$$ as a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk \cite{sfr}, the unit bidisk \cite{ando} and in the tetrablock \cite{sir and me1}. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a $Γ$-isometry. We compute its associated fundamental operator.

math.FA↗

arXiv2014

Explicit and unique construction of tetrablock unitary dilation in a certain case

Consider the domain $E$ in $\mathbb{C}^3$ defined by $$ E=\{(a_{11},a_{22},\text{det}A): A=\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\text{ with }\lVert A \rVert <1\}. $$ This is called the tetrablock. This paper constructs explicit boundary normal dilation for a triple $(A,B,P)$ of commuting bounded operators which has $\bar{E}$ as a spectral set. We show that the dilation is minimal and unique under a certain natural condition. As is well-known, uniqueness of minimal dilation usually does not hold good in several variables, e.g., Ando's dilation is not known to be unique. However, in the case of the tetrablock, the third component of the dilation can be chosen in such a way as to ensure uniqueness.

math.FA↗

arXiv2014

A Note on Tetrablock Contractions

A commuting triple of operators $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$ E = \{\underline{x}=(x_1,x_2,x_3)\in \mathbb{C}^3: 1-x_1z-x_2w+x_3zw \neq 0 \text{whenever}|z| \leq 1\text{and}|w| \leq 1 \} $$ is a spectral set. In this paper, we have constructed a functional model and produced a complete unitary invariant for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations $$ A-B^*P = D_PX_1D_P \text{and} B-A^*P=D_PX_2D_P, \text{where $X_1,X_2 \in \mathcal{B}(\mathcal{D}_P)$}, $$ play a big role. As a corollary to the functional model, we show that every pure tetrablock isometry $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is unitarily equivalent to $(M_{G_1^*+G_2z}, M_{G_2^*+G_1z},M_z)$ on $H^2_{\mathcal{D}_{P^*}}(\mathbb{D})$, where $G_1$ and $G_2$ are the fundamental operators of $(A^*,B^*,P^*)$. We prove a Beurling-Lax-Halmos type theorem for a triple of operators $(M_{F_1^*+F_2z},M_{F_2^*+F_1z},M_z)$, where $\mathcal{E}$ is a Hilbert space and $F_1,F_2 \in \mathcal{B}(\mathcal{E})$. We deal with a natural example of tetrablock contraction on functions space to find out its fundamental operators.

math.FA↗

arXiv2014

Admissible fundamental operators

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $Γ$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and $G$ is the fundamental operator of $(S^*,P^*)$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^*,P^*)$ respectively. Theorem 2 shows that this necessary condition is sufficient too provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for $Γ$-contractions are then applied to tetrablock contractions to figure out when two pairs $(F_1, F_2)$ and $(G_1, G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A, B, P)$ and its adjoint $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4.

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