ChapterZero » Adjoints (in Hilbert spaces)

Adjoints (in Hilbert spaces)

Mathematics — Alex @ 3:47 pm

I finally have a good grasp of adjoints, and I need to record it for the next time I need it. Let $\varphi$ be a bounded bilinear form, i.e. $\varphi : H \times H \rightarrow R$ , is linear in the first variable, conjugate linear in the second, and there is an M >0 such that $| \varphi(x,y) | \leq M \|x\| \|y\|$ . Then by the Riesz representation theorem, there is a bounded operator such that $\varphi(x,y) = \langle Ax, y \rangle$ .

Now, define a new bilinear form $\tilde \varphi (x,y) = \langle x, Ay \rangle$ . Then $\|\tilde \varphi(x,y) \| \leq |\langle x, Ay \rangle| \leq \|x\|\|Ay\| \leq \|A\|\|x\|\|y\|$ . Therefore, $\tilde \varphi$ is a bounded bilinear form, so there is a $A^\tilde$ such that $\varphi(x,y) = \langle x, Ay \rangle = \langle A^\star x, y \rangle$ . We call $A^\star$ the adjoint of .

Note immediately that $(A^\star)^\star = A$ , or in fancy language, $\star$ is involutive. Together with the fact $\|A^\star\| \leq \|A\|$ (from the last inequality above), this shows $\|A^\star\| = \|A\|$ .