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Gadadhar Misra

Publications and source records attributed to Gadadhar Misra.

At least 37 records · Page 2Linked to original sources
arXiv2016

Contractivity and complete contractivity for finite dimensional Banach Spaces

Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$ and let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contractive linear maps are always completely contractive thus remains open. We answer this question for balls of the form $Ω_\mathbf A$ in $\mathbb C^2.$

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arXiv2015

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 group on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $\mathbb C^2$ this condition is always satisfied. As an application we show that all homogeneous pairs of Cowen-Douglas operators are similar to direct sums of certain basic pairs.

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arXiv2015

Contractivity, complete contractivity and curvature inequalities

For any bounded domain $Ω$ in $\mathbb C^m,$ let ${\mathrm B}_1(Ω)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $\boldsymbol T$ in the Cowen-Douglas class ${\mathrm B}_1(Ω),$ let $N_{\boldsymbol T}(w)$ denote the restriction of $\boldsymbol T$ to the subspace ${\cap_{i,j=1}^m\ker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{\boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $ρ_{_{\!N_{\boldsymbol T}(w)}}(p) = p\big (N_{\boldsymbol T}(w) \big),\, p\in P[z_1, ..., z_m].$ We study the contractivity and complete contractivity of the homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}.$ Starting from the homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}},$ we construct a natural class of homomorphism $ρ_{_{\!N^{(λ)}(w)}}, λ>0,$ and relate the properties of $ρ_{_{\!N^{(λ)}(w)}}$ to that of $ρ_{_{\!N_{\boldsymbol T}(w)}}.$ Explicit examples arising from the multiplication operators on the Bergman space of $Ω$ are investigated in detail. Finally, it is shown that contractive properties of $ρ_{_{\!N_{\boldsymbol T}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{\boldsymbol T}$.

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arXiv2014

Curvature and the Second fundamental form in classifying quasi-homogeneous holomorphic curves and operators in the Cowen-Douglas class

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen-Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen-Douglas class obtained recently by the last author and A. Korányi. Specifically, the complex geometric invariants used for our classification are the curvature and the second fundamental forms inherent in the definition of a quasi-homogeneous operator. We show that these operators are irreducible and determine when they are strongly irreducible. Applications include the equality of the topological and algebraic K-group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos on similarity for these operators.

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arXiv2014

Rigidity of the flag structure for a class of Cowen-Douglas operators

The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid, that is, the unitary equivalence class of the operator and the flag structure determine each other. A complete set of unitary invariants, which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class, are obtained.

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arXiv2014

Flag structure for operators in the Cowen-Douglas class

The explicit description of homogeneous operators and localization of a Hilbert module naturally leads to the definition of a class of Cowen-Douglas operators possessing a flag structure. These operators are irreducible. We show that the flag structure is rigid in the sense that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.

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arXiv2012

Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class

The curvature $\mathcal K_T(w)$ of a contraction $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ is bounded above by the curvature $\mathcal K_{S^*}(w)$ of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this note, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle $E_T$ corresponding to the operator $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class $B_1(Ω)$, for a bounded domain $Ω$ in $\mathbb C^m$.

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arXiv2011

Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.

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arXiv2010

A classification of homogeneous operators in the Cowen-Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc $\mathbb D$ is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over $\mathbb D$, the ones corresponding to operators in the Cowen-Douglas class ${\mathrm B}_n(\mathbb D)$ are identified. The classification of homogeneous operators in ${\mathrm B}_n(\mathbb D)$ is completed using an explicit realization of these operators. We also show how the homogeneous operators in ${\mathrm B}_n(\mathbb D)$ split into similarity classes.

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arXiv2010

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z, w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S^{-1}(z, w) k(z, w) is a positive kernel function, where S(z, w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale \cite{Ath} and Ambrozie, Englis and Müller. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in {C}^m. Some consequences of this more general result are then explored in the case of several natural function algebras.

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arXiv2010

Resolution of singularities for a class of Hilbert modules

A short proof of the "Rigidity theorem" using the sheaf theoretic model for Hilbert modules over polynomial rings is given. The joint kernel for a large class of submodules is described. The completion $[\mathcal I]$ of a homogeneous (polynomial) ideal $\mathcal I$ in a Hilbert module is a submodule for which the joint kernel is shown to be of the form $$ \{p_i(\tfrac{\partial}{\partial \bar{w}_1}, ..., \tfrac{\partial}{\partial \bar{w}_m}) K_{[\mathcal I]}(\cdot,w)|_{w=0}, 1 \leq i \leq n\}, $$ where $K_{[\mathcal I]}$ is the reproducing kernel for the submodule $[\mathcal I]$ and $p_1, ..., p_n$ is some minimal "canonical set of generators" for the ideal $\mathcal I$. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A set of easily computable invariants for these submodules, using the monoidal transformation, are provided. Several examples are given to illustrate the explicit computation of these invariants.

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arXiv2009

Unitary invariants for Hilbert modules of finite rank

A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for which the localizations are of finite (but not constant) dimension, of an analytic function space with a reproducing kernel. The correspondence between analytic Hilbert modules of constant rank and holomorphic Hermitian bundles on domains of $\mathbb C^n$ due to Cowen and Douglas, as well as a natural analytic localization technique derived from the Hochschild cohomology of topological algebras play a major role in the proofs. A series of concrete computations, inspired by representation theory of linear groups, illustrate the abstract concepts of the paper.

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arXiv2009

A classification of homogeneous operators in the Cowen-Douglas class

A complete list of homogeneous operators in the Cowen-Douglas class $B_n(D)$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.

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arXiv2009

Multiplicity-free homogeneous operators in the Cowen-Douglas class

In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\mathbb D$ which are {\em homogeneous} with respect to the action of the group Möb -- the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. The {\em associated representation} for each of these operators is {\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the examples constructed previously.

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arXiv2006

Curvature calculations for a class of homogeneous operators

For an operator $T$ in the class ${\mathrm B}_n(Ω)$, introduced in \cite{CD}, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle $E_T$ determine the unitary equivalence class of the operator $T$. In the paper \cite{CD2}, the authors ask if there exists some pair of inequivalent oprators $T_1$ and $T_2$ for which the simultaneous unitary equivalence class of the curvature along with all the covariant derivatives coincide except for the derivative of the highest order. Here we show that some of the covariant derivatives are necessary to determine the unitary equivalence class of the operators in ${\mathrm B}_n(Ω)$. Our examples consist of homogeneous operators. For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives are determined from the simultaneous unitary equivalence class of these at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples.

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arXiv2006

On the irreducibility of a class of homogeneous operators

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra $\mathcal{A}(\mathbb D)$ as quotients of certain natural modules over the function algebra $\mathcal{A}(\mathbb D^2)$. These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class ${\mathrm B}_k(\mathbb D)$ and that they are mutually inequivalent, irreducible and homogeneous.

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arXiv2006

Equivalence of quotient Hilbert modules -- II

For any open, connected and bounded set $Ω\subseteq \mathbb C^m$, let $\mathcal A$ be a natural function algebra consisting of functions holomorphic on $Ω$. Let $\mathcal M$ be a Hilbert module over the algebra $\mathcal A$ and $\mathcal M_0\subseteq \mathcal M$ be the submodule of functions vanishing to order $k$ on a hypersurface $\mathcal Z \subseteq Ω$. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module $\mathcal Q = \mathcal M \ominus \mathcal M_0$ in the case of $k=2$. In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.

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