In this paper we study the generation of analytic semigroup in the space $L^2(\mathbb R^d)$, and the characterization of the domain, for a family of degenerate elliptic operators with unbounded coefficients, which includes some well-known operators arising in Mathematical Finance. To prove the generation of analytic semigroups, the operators of the family are assumed to satisfy suitable growth and compensation conditions. Under stronger assumptions, we obtain also the characterization of the domain. Finally, various consequences of the obtained results are considered in connection with some applications (see, e.g.,  for financial applications). In a forthcoming paper, part II of this work, we shall examine the problem of the generation of analytic semigroup in $L^p((\mathbb R^d)$, where $p\in(2,+\infty]$, and the characterization of the domain, for the same family of operators.
"Generation of analytic semigroups and domain characterization for degenerate elliptic operators with unbounded coefficients arising in financial mathematics. I." Differential Integral Equations 15 (9) 1085 - 1128, 2002. https://doi.org/10.57262/die/1356060765