# Volterra operator

A completely-continuous linear operator $V$ (cf. Completely-continuous operator), acting on a Banach space, whose spectrum consists of the point zero only. An example of a linear Volterra integral operator on the space of functions which are square-summable on $[a,b]$ is

$$V\phi(x)=\int\limits_a^x K(x,s)\phi(s)ds.$$

A non-linear Volterra integral operator is an operator of the form

$$V\phi(x)=\int\limits_a^x K(x,s,\phi(s))ds.$$

Named after V. Volterra, who studied the Volterra integral equations corresponding to such operators (cf. Volterra equation).