Norm of Self-Adjoint Operator

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a self-adjoint operator.


Then the norm of $A$ satisfies:

$\left\Vert{A}\right\Vert = \sup \left\{{ \left\vert{ \left\langle{Ah, h}\right\rangle_H }\right\vert: h \in H, \left\Vert{h}\right\Vert_H = 1 }\right\}$


Corollary

Suppose also that:

$\forall h \in H: \sequence {A h, h}_H = 0$



Then $A$ is the zero operator $\mathbf 0$.


Proof


Sources