# Manchester Geometry Seminar 2011/2012

Thursday 9 February 2012. The Frank Adams Room (Room 1.212), the Alan Turing Building. 3.15pm

## Operator Pencil Passing through a Given Operator Hovhannes Khudaverdian (University of Manchester)

khudian@manchester.ac.uk

Let $L_n$ be an operator of order $n$ acting on the space $\mathcal F_{\lambda_0}(M)$ of densities of weight $\lambda_0$ on a manifold $M$. The following question arises: Does there exist a pencil of operators (parametrized by weight $\lambda$) of order $n$ passing through a given operator $L_n$ and obeying some natural conditions? How many such pencils do exist? We give a precise formulation of this question and answer it in the case when $M$ is the line. We compare these answers with the following result: let $L$ and $L'_n$ be two operators of order $n=2$ acting on the spaces $\mathcal F_{\lambda_0}(M)$ and $\mathcal F_{\lambda'_0}(M)$ respectively. Then for almost all values of $\lambda_0$ and $\lambda'_0$ there exists a unique operator pencil of order $2$ passing through these operators. (Duval and Ovsienko, 1997; Kh. and Voronov, 2003).

The talk is based on my joint work with A. Biggs