I am working with the book "Introductory Functional Analysis With Applications", written by Kreyzig and I got the trouble with problem 1 and problem 2 in section 7.3, which are:
Problem 1:
"Let $X=C[0;1]$ and define $T: X \rightarrow X$ by $Tx=vx$, where $v \in X$ is fixed. Find $\sigma (T)$. Note that $\sigma (T)$ is closed. "
Problem 2:
"Find a linear operator $T: C[0;1] \rightarrow C[0;1]$ whose spectrum is a given interval $[a,b]$."
There is a hint for problem 1 that $\sigma (T)$ is the range of $v$. However, I have no idea of proving this.
Could you please help me to explain those problems in detail?
Thank you so much.
If $\lambda = v(t_0)$ for some $t_0\in [0,1]$, then $(T-\lambda I)x$ vanishes at $t_0$ for every $x\in C[0,1]$, which proves that $T-\lambda I$ is not surjective if $\lambda$ is in the range of $v$. So the range of $v$ is contained in the spectrum of $T$. If $\lambda$ is not in the range of $v$, then $\inf_{t\in [0,1]}|\lambda-v(t)| = \epsilon > 0$ because the range is compact, which gives the continuous invertibility of multiplication by $T-\lambda I$: $$ (T-\lambda I)^{-1}x = \frac{1}{v-\lambda}x. $$ So $\sigma(T)=\{ v(t) : t \in [0,1] \}$.