This is Rudin's functional Analysis chapter $10$, exercise $13$.
I am confused about the notation $\sigma_A(f)$, what does that mean?(What role does the subscript $A$ play here). And can someone illustrate how to solve this question?
Update: thanks to the comment, and a theorem in Rudin:
$B$ is the collection of polynomials of $x$ ($|x| \in [1,2]$), whose power ranges from negative to positive. So the only possible $\lambda$ (here I use $\lambda$ to denote the spectral, while $x$ as variable in the function, not the way in the question) is 1.
Actually, if we want $(\lambda*e-f)(x) *P(x)=1$ when $\lambda$ is not $1$, we can simply choose $P(x)$ to be $\frac{1}{(\lambda-1)x}$.
So by the theorem in Rudin, $\sigma_A=\sigma_B={1}$.
Is this correct?