“Friedrichs extension”的意思、由来-开放百科全书

词条

Friedrichs extension

释义

Examples
Definition of Friedrichs extension
Krein's theorem on non-negative self-adjoint extensions
See also
Notes
References

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

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

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 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

is a sesquilinear form on dom T and

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

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

Define an operator A by

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

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

T₁ is the Friedrichs extension of 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

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.

Notes

References

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

1 : Operator theory

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Examples

Definition of Friedrichs extension

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

See also

Notes

References