PhD Thesis colloquium

Title: Homogeneous operators
Speaker: Somnath Hazra (IISc Mathematics)
Date: 19 June 2017
Time: 2 pm
Venue: LH-1, Mathematics Department

It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form

where, $\sigma_{L}$ and $\sigma_{R}$ are projective representation of the Mobius group Mob with a common multiplier. We give another proof of the ``product formula’’. Also, we prove that the projective representations $\sigma_L$ and $\sigma_R$ for a class of multiplication operators, the two representations $\sigma_{R}$ and $\sigma_{L}$ are unitarily equivalent to certain known pair of representations $\sigma_{\lambda + 1}$ and $\sigma_{\lambda - 1},$` respectively. These are described explicitly.

Let $G$ be either (i) the direct product of $n$-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc $\mathbb D^n.$ A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots ,T_n)$ is said to be $G$-homogeneous if the joint spectrum of $\mathsf{T}$ lies in $\widebar{\mathbb{D}}^n$ and $\varphi(\mathsf{T}),$ defined using the usual functional calculus, is unitarily equivalent with $\mathsf{T}$ for all $\varphi \in G.$

We show that a commuting tuple $\mathsf{T}$ in the Cowen-Douglas class of rank $1$ is $G$ - homogeneous if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n} \frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n} \frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda}},$ where $\lambda,$ $\lambda_i$, $1 \leq i \leq n,$ are positive real numbers, according as $G$ is as in (i) or (ii).

Let $\mathsf T:=(T_1, \ldots ,T_{n-1})$ be a $G$-homogeneous $(n-1)$-tuple of rank $1$ Cowen-Douglas class, where $G$ is the the direct product of $n-1$-copies of the bi-holomorphic automorphism group of the disc. Let $\hat{T}$ be an irreducible homogeneous (with respect to the bi-holomorphic group of automorphisms of the disc) operator in the Cowen-Douglas class on the disc of rank $2$. We show that every irreducible $G$ - homogeneous operator, $G$ as in (i), of rank $2$ must be of the form

We also show that if $G$ is chosen to be the group as in (ii), then there are no irreducible $G$- homogeneous operators of rank $2.$


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Last updated: 17 Oct 2019