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Gadadhar Misra

Publications and source records attributed to Gadadhar Misra.

At least 19 recordsLinked to original sources
arXiv2026

Extreme points, positive Grothendieck constants and tensor product norms

We study several interrelated problems arising from the interplay between extreme point theory, Grothendieck-type inequalities, and tensor product norms. We develop a general framework for characterizing the extreme points of the set of positive contractions $\mathcal{A}_{X\to Y}$ between finite-dimensional Banach spaces, with explicit results for $X=\ell_1^n$, $Y=\ell_\infty^n$ and vice versa. These characterizations are applied to evaluate several constants exactly. We show that the positive Grothendieck constant $K_G^{+,\mathbb{R}}(3)$ equals $9/8$ and that the smallest constant $ρ^{+}(X)$ for which $\|A\|_π\leqslant ρ^{+}(X)\|A\|_ε$ holds for all $A \geqslant 0$ equals $5/4$ when $X=\ell^3_\infty(\mathbb{R})$. We also prove that $ρ^+(X)=1$ when $X=\ell_\infty^n(\mathbb{C})$ and $n\leqslant 3$. Finally, we prove that $ρ^+(X) = 1$ for every 2-dimensional subspace $X$ of $\ell^3_\infty(\mathbb{C})$; since this is stronger than the 2-summing property, it recovers Proposition~4.4 of \cite{AFJS95}.

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arXiv2025

On a variant of the Grothendieck inequality and estimates on tensor product norms

We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C >0$ such that $\|A\otimes \operatorname{id}_{F}\|_{E_m\check{\otimes}F\to E_n^*\hat{\otimes}F}\leqslant C \|A\|_{E_m\to E_n^*}$ for all $m,n\in\mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^*$ must have finite cotype. Moreover, assuming that $F$ has the bounded approximation property and that the conjecture in \cite{PisierDuality} has an affirmative answer, we show that $(E_n^*)_{n\geqslant 1}$ satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces $(E,F)$ is closely related to another interesting quantity introduced recently in \cite{XOR games and GT} comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces $E$ and $F$. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive \emph{little} Parrott homomorphisms $\varrho_V : H^\infty(Ω) \to M_{n}$, where $Ω$ is the dual unit ball of a finite dimensional Banach space $(E,\|\cdot\|)$, we prove the sharp estimate $ \|\varrho_V\|_{\mathrm{cb}}\leq\sqrt{γ(E)}, $ $γ(E)$ being the positive Grothendieck constant associated with the pair $(E, \ell^n_2)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^+(\ell_\infty^4,\ell_2^2) \geq 1.1658$ obtained in this paper.

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arXiv2025

Homogeneous analytic Hilbert modules -- the case of non-transitive action

This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $Ω\subset \mathbb{C}^m$ that are homogeneous under the natural action of the group $G$. In a departure from the past studies of such questions, here we don't assume transitivity of the group action. The primary finding reveals that unitary invariants such as curvature and the reproducing kernel of a homogeneous analytic Hilbert module can be deduced from their values on a fundamental set $Λ$ of the group action. Next, utilizing these techniques, we examine the analytic Hilbert modules associated with the symmetrized bi-disc $\mathbb{G}_2$ and its homogeneity under the automorphism group of $\mathbb{G}_2$. It follows from one of our main theorems that none of the weighted Bergman metrics on the symmetrized bi-disc is Kähler-Einstein.

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arXiv2024

Mackey Imprimitivity and commuting tuples of homogeneous normal operators

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting $d$- tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a $*$- algebra representation $ρ$ of $C_0(\mathbb{S})$ (where $\mathbb S$ is a locally compact Hausdorff space) into a direct sum. If there is a group $G$ acting transitively on $\mathbb{S}$ and is adapted to the $*$- representation $ρ$ via a unitary representation $U$ of the group $G$, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation $U$ becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space $S\subset \mathbb {C}^d$ decomposes into finitely many $G$- orbits. In such cases, the imprimitivity based on $S$ admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of $G$- orbits.

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arXiv2023

Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

Let $\mathcal U(d)$ be the group of $d\times d$ unitary matrices. We find conditions to ensure that a $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol T$ is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq \mbox{\rm Hol}(\mathbb B_d, \mathbb C^n)$, $n= \dim \cap_{j=1}^d \ker T^*_{j}.$ We describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative kernels $K$ quasi-invariant under the action of $\mathcal U(d)$. We classify quasi-invariant kernels $K$ transforming under $\mathcal U(d)$ with two specific choice of multipliers. A crucial ingredient of the proof is that the group $SU(d)$ has exactly two inequivalent irreducible unitary representations of dimension $d$ and none in dimensions $2, \ldots , d-1$, $d\geq 3$. We obtain explicit criterion for boundedness, reducibility and mutual unitary equivalence among these operators.

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arXiv2022

Geometric invariants for a class of submodules of analytic Hilbert modules via the sheaf model

Let $Ω\subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(Ω)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\boldsymbol{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\boldsymbol{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\boldsymbol{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. The sheaf $\mathcal S^\mathcal H$ associated to analytic Hilbert module $\mathcal H$ is the sheaf $\mathcal O(Ω)$ of holomorphic functions on $Ω$ and hence is free. However, the subsheaf $\mathcal S^{\mathcal [\mathcal I]}$ associated to $[\mathcal I]$ is coherent and not necessarily locally free. Building on the earlier work of \cite{BMP}, we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set $V_{[\mathcal I]}$ is a submanifold of codimension $t$, then there is a unique local decomposition for the kernel $K_{[\mathcal I]}$ along the zero set that serves as a holomorphic frame for a vector bundle on $V_{[\mathcal I]}$. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule $[\mathcal I] \subseteq \mathcal H$.

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arXiv2022

The relationship of the Gaussian curvature with the curvature of a Cowen-Douglas operator

It has been recently shown that if $K$ is a sesqui-analytic scalar valued non-negative definite kernel on a domain $Ω$ in $\mathbb C^m$, then the function $\big(K^2\partial_i\bar{\partial}_j\log K\big )_{i,j=1}^ m,$ is also a non-negative definite kernel on $Ω$. In this paper, we discuss two consequences of this result. The first one strengthens the curvature inequality for operators in the Cowen-Douglas class $B_1(Ω)$ while the second one gives a relationship of the reproducing kernel of a submodule of certain Hilbert modules with the curvature of the associated quotient module.

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arXiv2021

A trace inequality for commuting tuple of operators

For a commuting $d$- tuple of operators $\boldsymbol T$ defined on a complex separable Hilbert space $\mathcal H$, let $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ be the $d\times d$ block operator $\big (\!\!\big (\big [ T_j^* , T_i\big ]\big )\!\!\big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, \ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ must be $0$. We show that if the $d$- tuple $\boldsymbol T$ is cyclic, the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $\mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $\big [\!\! \big [ \boldsymbol T^* , \boldsymbol T\big ] \!\!\big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.

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arXiv2019

Toeplitz $C^*$-Algebras on Boundary Orbits of Symmetric Domains

We study Toeplitz operators on Hilbert spaces of holomorphic functions on symmetric domains, and more generally on certain algebraic subvarieties, determined by integration over boundary orbits of the underlying domain. The main result classifies the irreducible representations of the Toeplitz $C^*$-algebra generated by Toeplitz operators with continuous symbol. This relies on the limit behavior of "hypergeometric" measures under certain peaking functions.

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arXiv2019

A product formula for homogeneous characteristic functions

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $φ(T)$ is unitarily equivalent to $T$ for all $φ$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $σ$ of Möb is said to be associated with an operator T if $φ(T)= σ(φ)^\star T σ(φ)$ for all $φ$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $σ$, then there is a unique projective unitary representation $\hatσ$, extending $σ$, associated with the minimal unitary dilation of $T$. The representation $\hatσ$ is given in terms of $σ$ by the formula $$ \hatσ = (π\otimes D_1^+) \oplus σ\oplus (π_\star \otimes D_1^-), $$ where $D_1^\pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(\mathbb D)$, and $π, π_\star$ are representations of Möb living on the two defect spaces of $T$ defined explicitly in terms of $σ$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $θ_T$ has the product form $θ_T(z) = π_\star(φ_z)^* θ_T(0) π(φ_z),$ $z\in \mathbb D$, where $φ_z$ is the involution in Möb mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $π_\star$ and $π$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.

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arXiv2019

Decomposition of the tensor product of two Hilbert modules

Given a pair of positive real numbers $α, β$ and a sesqui-analytic function $K$ on a bounded domain $Ω\subset \mathbb C^m$, in this paper, we investigate the properties of the sesqui-analytic function $\mathbb K^{(α, β)}:= K^{α+β}\big(\partial_i\bar{\partial}_j\log K\big )_{i,j=1}^ m,$ taking values in $m\times m$ matrices. One of the key findings is that $\mathbb K^{(α, β)}$ is non-negative definite whenever $K^α$ and $K^β$ are non-negative definite. In this case, a realization of the Hilbert module determined by the kernel $\mathbb K^{(α,β)}$ is obtained. Let $\mathcal M_i$, $i=1,2,$ be two Hilbert modules over the polynomial ring $\mathbb C[z_1, \ldots, z_m]$. Then $\mathbb C[z_1, \ldots, z_{2m}]$ acts naturally on the tensor product $\mathcal M_1\otimes \mathcal M_2$. The restriction of this action to the polynomial ring $\mathbb C[z_1, \ldots, z_m]$ obtained using the restriction map $p \mapsto p_{|Δ}$ leads to a natural decomposition of the tensor product $\mathcal M_1\otimes \mathcal M_2$, which is investigated. Two of the initial pieces in this decomposition are identified.

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arXiv2019

Singular Hilbert modules on Jordan-Kepler varieties

We study submodules of analytic Hilbert modules defined over certain algebraic varieties in bounded symmetric domains, the so-called Jordan-Kepler varieties $V_\ell$ of arbitrary rank $\ell.$ For $\ell>1$ the singular set of $V_\ell$ is not a complete intersection. Hence the usual monoidal transformations do not suffice for the resolution of the singularities. Instead, we describe a new higher rank version of the blow-up process, defined in terms of Jordan algebraic determinants, and apply this resolution to obtain the rigidity of the submodules vanishing on the singular set.

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arXiv2019

Operators in the Cowen-Douglas class and related topics

Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although the study of self-adjoint operators goes back a few decades, the non self-adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like $\mathcal A(Ω)$, where $Ω\subseteq \mathbb C^m$ is a bounded domain, consisting of complex valued functions which are holomorphic on some open set $U$ containing $\overlineΩ$, the closure of $Ω$. The book, ''Hilbert Modules over function algebra'', R. G. Douglas and V. I. Paulsen showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The book, ''Spectral decomposition of analytic sheaves'', J. Eschmeier and M. Putinar and the book, ''Analytic Hilbert modules'', X. Chen and K. Guo, provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.

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arXiv2018

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous $n$-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in $\mathbb C^n$.

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arXiv2018

Curvature Inequalities and Extremal Operators

A curvature inequality is established for contractive commuting tuples of operators in the Cowen-Douglas class of rank n. Properties of the extremal operators, that is, the operators which achieve equality, are investigated. Specifically, a substantial part of a well known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.

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arXiv2017

The Carathéodory-Fejér interpolation problem for the polydisc

We give an algorithm for finding a solution to the Carathéodory-Fejér interpolation problem on the polydisc $\mathbb D^n,$ whenever it exists. A necessary condition for the existence of a solution becomes apparent from this algorithm. A generalization of the well-known theorem due to Nehari has been obtained. A proof of the Korányi--Pukánszky theorem is given using the spectral theorem.

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arXiv2017

Reducing sub-modules of the Bergman module $\mathbb A^{(λ)}(\mathbb D^n)$ under the action of the symmetric group

The weighted Bergman spaces on the polydisc, $\mathbb A^{(λ)}(\mathbb D^n)$, $λ>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $χ_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $χ_{\boldsymbol p}(1) \ne χ_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.

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