Symmetric operator

Summary

A symmetric operator is a densely defined linear operator $T$ on a Hilbert space with $\ip{Tx}{y} = \ip{x}{Ty}$ for all $x, y$ in its domain. For bounded operators defined on the whole space, symmetric is the same as self-adjoint. For unbounded operators, such as differential operators, symmetry is weaker than self-adjointness. The gap between the two is the subject of the theory of self-adjoint extensions.