# Friedrichs extension

In functional analysis, the **Friedrichs extension** is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show.

An operator *T* is non-negative if

## Examples[edit]

**Example**. Multiplication by a non-negative function on an *L*^{2} space is a non-negative self-adjoint operator.

**Example**. Let *U* be an open set in **R**^{n}. On *L*^{2}(*U*) we consider differential operators of the form

where the functions *a*_{i j} are infinitely differentiable real-valued functions on *U*. We consider *T* acting on the dense subspace of infinitely differentiable complex-valued functions of compact support, in symbols

If for each *x* ∈ *U* the *n* × *n* matrix

is non-negative semi-definite, then *T* is a non-negative operator. This means (a) that the matrix is hermitian and

for every choice of complex numbers *c*_{1}, ..., *c*_{n}. This is proved using integration by parts.

These operators are elliptic although in general elliptic operators may not be non-negative. They are however bounded from below.

## Definition of Friedrichs extension[edit]

The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces.
If *T* is non-negative, then

is a sesquilinear form on dom *T* and

Thus Q defines an inner product on dom *T*. Let *H*_{1} be the completion of dom *T* with respect to Q. *H*_{1} is an abstractly defined space; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom *T*. It is not obvious that all elements in *H*_{1} can be identified with elements of *H*. However, the following can be proved:

The canonical inclusion

extends to an *injective* continuous map *H*_{1} → *H*. We regard *H*_{1} as a subspace of *H*.

Define an operator *A* by

In the above formula, *bounded* is relative to the topology on *H*_{1} inherited from *H*. By the Riesz representation theorem applied to the linear functional φ_{ξ} extended to *H*, there is a unique *A* ξ ∈ *H* such that

**Theorem**. *A* is a non-negative self-adjoint operator such that *T*_{1}=*A* - I extends *T*.

*T*_{1} is the Friedrichs extension of *T*.

## Krein's theorem on non-negative self-adjoint extensions[edit]

M. G. Krein has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator *T*.

If *T*, *S* are non-negative self-adjoint operators, write

if, and only if,

**Theorem**. There are unique self-adjoint extensions *T*_{min} and *T*_{max} of any non-negative symmetric operator *T* such that

and every non-negative self-adjoint extension *S* of *T* is between *T*_{min} and *T*_{max}, i.e.

## See also[edit]

## Notes[edit]

## References[edit]

- N. I. Akhiezer and I. M. Glazman,
*Theory of Linear Operators in Hilbert Space*, Pitman, 1981.