Hellinger-Toeplitz's theorem

The Hellinger-Toeplitz theorem is a mathematical theorem from functional analysis . It is named after the mathematicians Ernst Hellinger and Otto Toeplitz .

formulation

Let there be a Hilbert space and a symmetric linear operator , that is, an operator which for all the equation ${\displaystyle H}$ ${\displaystyle T:H\rightarrow H}$ ${\displaystyle x,\,y\in H}$

${\displaystyle \langle Tx,y\rangle =\langle x,Ty\rangle }$

Fulfills. Then is steady. ${\displaystyle T}$

proof

According to the closed graph theorem , it is sufficient to show that if is a null sequence and convergent, then is . Using the continuity of the inner product on and is then followed ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle Tx_{n}}$ ${\displaystyle \lim _{n\rightarrow \infty }Tx_{n}=0}$
${\displaystyle H}$ ${\displaystyle y:=\lim _{n\rightarrow \infty }Tx_{n}}$

${\displaystyle \langle y,y\rangle =\langle \lim _{n\rightarrow \infty }Tx_{n},y\rangle =\lim _{n\rightarrow \infty }\langle Tx_{n},y\rangle =\lim _{n\rightarrow \infty }\langle x_{n},Ty\rangle =\langle \lim _{n\rightarrow \infty }x_{n},Ty\rangle =\langle 0,Ty\rangle =0,}$

so . ${\displaystyle y=0}$

Inferences

• Since the operator is linear and continuous, it is also bounded. ${\displaystyle T}$
• Every symmetric operator defined everywhere on is self-adjoint . ${\displaystyle H}$
• Unbounded self-adjoint operators can at most be defined on a dense subset of a Hilbert space.

generalization

One can weaken the condition in Hellinger-Toeplitz's theorem:

Let and Hilbert spaces and a linear operator that has an adjoint , that is: There is an operator that is for all and the equation ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle T:H_{1}\rightarrow H_{2}}$ ${\displaystyle S:H_{2}\rightarrow H_{1}}$ ${\displaystyle x\in H_{1}}$ ${\displaystyle y\in H_{2}}$

${\displaystyle \langle Tx,y\rangle _{H_{2}}=\langle x,Sy\rangle _{H_{1}}}$

Fulfills. Then and are steady. ${\displaystyle T}$ ${\displaystyle S}$

The proof is analogous.

literature

• Dirk Werner : Functional Analysis (Springer, 5th edition 2005)