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

${\displaystyle \langle \xi \mid T\xi \rangle \geq 0\quad \xi \in \operatorname {dom} \ T}$

Examples

Example. Multiplication by a non-negative function on an L² space is a non-negative self-adjoint operator.

Example. Let U be an open set in Rⁿ. On L²(U) we consider differential operators of the form

${\displaystyle [T\phi ](x)=-\sum _{i,j}\partial _{x_{i}}\{a_{ij}(x)\partial _{x_{j}}\phi (x)\}\quad x\in U,\phi \in \operatorname {C} _{c}^{\infty }(U),}$

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

${\displaystyle \operatorname {C} _{c}^{\infty }(U)\subseteq L^{2}(U).}$

If for each x ∈ U the n×n matrix

${\displaystyle {\begin{bmatrix}a_{11}(x)&a_{12}(x)&\cdots &a_{1n}(x)\\a_{21}(x)&a_{22}(x)&\cdots &a_{2n}(x)\\\vdots &\vdots &\ddots &\vdots \\a_{n1}(x)&a_{n2}(x)&\cdots &a_{nn}(x)\end{bmatrix}}}$

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

${\displaystyle \sum _{i,j}a_{ij}(x)c_{i}{\overline {c_{j}}}\geq 0}$

for every choice of complex numbers c₁, ..., 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

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

${\displaystyle \operatorname {Q} (\xi ,\eta )=\langle \xi \mid T\eta \rangle +\langle \xi \mid \eta \rangle }$

is a sesquilinear form on dom T and

${\displaystyle \operatorname {Q} (\xi ,\xi )=\langle \xi \mid T\xi \rangle +\langle \xi \mid \xi \rangle \geq \|\xi \|^{2}.}$

Thus Q defines an inner product on dom T. Let H₁ be the completion of dom T with respect to Q. H₁ 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 H₁ can be identified with elements of H. However, the following can be proved:

The canonical inclusion

${\displaystyle \operatorname {dom} T\rightarrow H}$

extends to an injective continuous map H₁ → H. We regard H₁ as a subspace of H.

Define an operator A by

${\displaystyle \operatorname {dom} \ A=\{\xi \in H_{1}:\phi _{\xi }:\eta \mapsto \operatorname {Q} (\xi ,\eta ){\mbox{ is bounded linear.}}\}}$

In the above formula, bounded is relative to the topology on H₁ 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

${\displaystyle \operatorname {Q} (\xi ,\eta )=\langle A\xi \mid \eta \rangle \quad \eta \in H_{1}}$

Theorem. A is a non-negative self-adjoint operator such that T₁=A - I extends T.

T₁ is the Friedrichs extension of T.

Another way to obtain this extension is as follows. Let :${\displaystyle L:H_{1}\rightarrow H}$ be the bounded inclusion operator. The inclusion is a bounded injective with dense image. Hence ${\displaystyle LL^{*}:H\rightarrow H}$ is a bounded injective operator with dense image, where ${\displaystyle L^{*}}$ is the adjoint of ${\displaystyle L}$ as an operator between abstract Hilbert spaces. Therefore the operator ${\displaystyle A:=(LL^{*})^{-1}}$ is a non-negative self-adjoint operator whose domain is the image of ${\displaystyle LL^{*}}$. Then ${\displaystyle A-I}$ extends T.

Krein's theorem on non-negative self-adjoint extensions

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

${\displaystyle T\leq S}$

if, and only if,

• ${\displaystyle \operatorname {dom} (S^{1/2})\subseteq \operatorname {dom} (T^{1/2})}$
• ${\displaystyle \langle T^{1/2}\xi \mid T^{1/2}\xi \rangle \leq \langle S^{1/2}\xi \mid S^{1/2}\xi \rangle \quad \forall \xi \in \operatorname {dom} (S^{1/2})}$

Theorem. There are unique self-adjoint extensions T_min and T_max of any non-negative symmetric operator T such that

${\displaystyle T_{\mathrm {min} }\leq T_{\mathrm {max} },}$

and every non-negative self-adjoint extension S of T is between T_min and T_max, i.e.

${\displaystyle T_{\mathrm {min} }\leq S\leq T_{\mathrm {max} }.}$

See also

• Energetic extension
• Extensions of symmetric operators

References

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