In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

The term "unbounded operator" can be misleading, since

• "unbounded" should sometimes be understood as "not necessarily bounded";
• "operator" should be understood as "linear operator" (as in the case of "bounded operator");
• the domain of the operator is a linear subspace, not necessarily the whole space;
• this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
• in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain.

The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible.