résolvent

In mathematics, a resolvent is a function associated with an operator, typically a linear operator on a Banach space or Hilbert space. The resolvent of an operator A is denoted by R(z, A) and is defined as the inverse of the operator (zI - A), where z is a complex number and I is the identity operator. So, R(z, A) = (zI - A)^-1.

The resolvent is defined for all complex numbers z for which the operator (zI - A) is invertible. The set of such z is called the resolvent set of A. The complement of the resolvent set in the complex plane is known as the spectrum of A. The resolvent function is analytic in its domain of definition, meaning it is differentiable in the complex sense.

The resolvent plays a crucial role in spectral theory. It provides a way to understand the properties of an operator by examining its behavior for different complex values of z. For instance, the singularities of the resolvent function precisely correspond to the spectrum of the operator. The resolvent can also be used to express functions of operators, such as e^At, which are important in the study of differential equations.

The resolvent satisfies several important identities, such as the first and second resolvent identities, which relate the resolvents of an operator at different points. These identities are fundamental tools for manipulating and analyzing resolvent expressions.