# Operator pencil passing through a given operator

A. Biggs, Hovhannes Khudaverdyan

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## Abstract

Let $\Delta$ be a linear differential operator acting onspace $\F_{\l_0}$ of densities of a given weight$\l_0$ on the manifold $M$.One can consider a pencil of operators $\Pi(L)=\{L_\l\}$passing through the operator $L$ such that any operator $\Delta_\l$is a linear differential operator acting on a space$\F_\l$ of densities of weight $\l$,where $\l$ is an arbitrary real number. A pencil $\{L_\l\}=\Pi(L)$can be identified with a linear differential operator actingon algebra of densitiesof all weights. The existence of invariant scalarproduct in the algebra of densities implies the natural decompositionof operators on algebra of densities, i.e. pencils of operatorson self-adjoint and antiself-adjoint operators. We study lifting maps $L\to \Pi(L)$ which take values in self-adjointor anti-self-adjoint operators, and on the other hand we study$G$-equivariant lifting mapsin the case if $G$ is a group of diffeomorphisms of $M$ orits subgroup which preserves a volume form on $M$ or a projective structureon $M$. We analyze in particular the cases where therestrictions above imply the exitence of unique lifting.
Original language English 27 Journal of Mathematical Physics 54 12 https://doi.org/10.1063/1.4839418 Published - 2013

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