Hellinger-Toeplitz theorem

The Hellinger – Toeplitz theorem is the result of a functional analysis establishing the boundedness of a symmetric operator in a Hilbert space .

Wording

Let be $H$ ${\ displaystyle H}$ $H$ - Hilbert space . If for a linear operator $A:\,H\to H$ ${\ displaystyle A: \, H \ to H}$ $A:\,H\to H$ there is a linear operator $B:\,H\to H$ ${\ displaystyle B: \, H \ to H}$ $B:\,H\to H$ satisfying the condition $(Ax,y)=(x,By)\;\forall x,y\in H$ ${\ displaystyle (Ax, y) = (x, By) \; \ forall x, y \ in H}$ $(Ax,y)=(x,By)\;\forall x,y\in H$ then the operator $A$ ${\ displaystyle A}$ $A$ is limited .

In particular, any symmetric operator defined on the whole space is bounded, that is, a linear operator satisfying the condition $(Ax,y)=(x,Ay)\;\forall x,y\in H$ ${\ displaystyle (Ax, y) = (x, Ay) \; \ forall x, y \ in H}$ $(Ax,y)=(x,Ay)\;\forall x,y\in H$ .

Remarks

An essential condition of the theorem is the condition for the definiteness of the operator on the entire Hilbert space .

Consequences

Every symmetric operator defined on the entire Hilbert space is self-adjoint .
A self-adjoint unbounded operator cannot be defined on the entire Hilbert space .

Hellinger-Toeplitz theorem

Wording

Remarks

Consequences

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