It is known that the characteristic function $\theta_T$ of a homogeneous contraction $T$ with an associated representation $\pi$ is of the form \begin{equation} \theta_T(a) = \sigma_{L}(\phi_a)^* \theta(0) \sigma_{R}(\phi_a), \end{equation}

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 `$\overline{\mathbb{D}}^n$`

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
\begin{equation}
(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes
I_{\widehat{H}}, I_H \otimes \hat{T}).
\end{equation}

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 $2.$

- All seminars.
- Seminars for 2017

Last updated: 29 Nov 2023