Are the creation operators on the fermionic Fock space bounded linear operators?

Question

Let $H$ be a Hilbert space, denoting single-particle states, and $\mathfrak{F}$ be the fermionic Fock space. If $f\in H$, then is the creation operator $c^*(f)$ a bounded linear operator on $\mathfrak{F}$? If so, what is its norm?

Answer 1

The operator is indeed bounded, its norm is $\|f\|$. A fun trick to see what the norm is can be done by utilising the CAR: $$\{c^*(f), c(f)\}=\|f\|^2, \qquad2c^*(f)\,c^*(f)=\{c^*(f),c^*(f)\}=0=\{c(f),c(f)\}=2c(f)\,c(f).$$ So you get: $$c^*(f)\,c(f)\,c^*(f)\,c(f)= c^*(f)\{c(f),c^*(f)\} c(f)-c^*(f)\,c^*(f)\,c(f)\,c(f)=\|f\|^2c^*(f)\,c(f).$$ Which you use for: $$\|c^*(f)\|^4 = (\|c^*(f)\|^2)^2 = \|c^*(f)c(f)\|^2 = \|c^*(f)c(f)c^*(f)c(f)\|=\|f\|^2\,\|c^*(f)\,c(f)\| \\ =\|f\|^2\,\|c^*(f)\|^2$$ from which $\|c^*(f)\|=\|f\|$ follows. (In the above line the relation $\|A\|^2=\|AA^*\|$ was used twice, once for $A=c^*(f)$ and once for $A=c^*(f)\,c(f)$.)