In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems.

This article discusses a few related problems of this type. The unifying theme is that each problem has an operator-theoretic characterization which gives a corresponding parametrization of solutions. More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.

Let ${\displaystyle H}$ be a Hilbert space. A linear operator ${\displaystyle A}$ acting on ${\displaystyle H}$ with dense domain ${\displaystyle \operatorname {dom} (A)}$ is symmetric if

If ${\displaystyle \operatorname {dom} (A)=H}$, the Hellinger-Toeplitz theorem says that ${\displaystyle A}$ is a bounded operator, in which case ${\displaystyle A}$ is self-adjoint and the extension problem is trivial. In general, a symmetric operator is self-adjoint if the domain of its adjoint, ${\displaystyle \operatorname {dom} (A^{*})}$, lies in ${\displaystyle \operatorname {dom} (A)}$.

When dealing with unbounded operators, it is often desirable to be able to assume that the operator in question is closed. In the present context, it is a convenient fact that every densely defined, symmetric operator ${\displaystyle A}$ is closable. That is, ${\displaystyle A}$ has the smallest closed extension, called the closure of ${\displaystyle A}$. This can be shown by invoking the symmetric assumption and Riesz representation theorem. Since ${\displaystyle A}$ and its closure have the same closed extensions, it can always be assumed that the symmetric operator of interest is closed.

In the next section, a symmetric operator will be assumed to be densely defined and closed.

If an operator ${\displaystyle A}$ on the Hilbert space ${\displaystyle H}$ is symmetric, when does it have self-adjoint extensions? An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of ${\displaystyle A}$) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all. Thus, we would like a classification of its self-adjoint extensions.

The first basic criterion for essential self-adjointness is the following: