I have following excercice:
Excercice (where H is a Hilbert-space) so far we have introduced the normal functional calculus, but not the holomorphic. my idea was to put $f_t(z):=e^{tz}$ and use this Lemma:
Lemma Now let R be close to ||A|| and proceeded: $\|f_t(A)\| = \| \frac{1}{2 \pi i} \int_{|w|=R} R_A(w) f_t(w) dw\| $ $\leq \frac{1}{2 \pi} 2 R \pi \sup_{z \in \delta B(0,R)} \|R_A(z) \| \sup_{z \in \delta B(0,R)} \| e^{tz}\| $
In the lecture we had $\|R_A(z)\|=\frac{1}{dist(z,(\sigma ))}$ and since R is close to ||A|| we can further estimate:
$\leq \sup_{z \in \delta B(0,R)}|e^{t re(z)}|$
now my idea was to let R go to ||A|| but I dont see how this helps. Am I on the right track? Or maybe using the series expansion would yield a better result?
kind regards