Exercise
Posted: Wed Mar 08, 2017 11:37 am
Using the lemma http://www.mathimatikoi.org/forum/viewt ... =28&t=1133" onclick="window.open(this.href);return false;
prove that if \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) is a Hilbert space and \(\displaystyle{U:H\to H}\)
is a \(\displaystyle{\mathbb{C}}\) -linear and bounded operator such that \(\displaystyle{||U||\leq 1}\),
with \(\displaystyle{F=\left\{h\in H\,,U(h)=h\right\}\,\,,N=\left\{U(h)-h\in H\,,h\in H\right\}}\) , then
\(\displaystyle{N=F^{\perp}}\).
prove that if \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) is a Hilbert space and \(\displaystyle{U:H\to H}\)
is a \(\displaystyle{\mathbb{C}}\) -linear and bounded operator such that \(\displaystyle{||U||\leq 1}\),
with \(\displaystyle{F=\left\{h\in H\,,U(h)=h\right\}\,\,,N=\left\{U(h)-h\in H\,,h\in H\right\}}\) , then
\(\displaystyle{N=F^{\perp}}\).