Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.
Operator theory is the study of linear operators between various (topological) vector spaces. The quintessential examples being differential and integral operators on function spaces over some domain, and hence operator theory is deeply connected to functional analysis. It has applications to and applications from differential equations, representation theory and mathematical physics.