The Spectrum of the derivative operator in a specific Banach space

Question

Consider the Banach space $X=\left\{u\in C^1([0,1]):\, u(0)=0\right\}$ and the subspace $D=\{u\in C^2([0,1]):\, u(0)=u(1)=u'(0)=0\}$, and the operator $A:D\longrightarrow X$ defined by $Au=u'$. I have made attempts to compute the spectrum and the resolvent operator (if it exists) for $A$ by solving the equation $$\lambda u-u'=f,$$ for some $f\in X$, but I have not succeeded. I would greatly appreciate it if someone could provide assistance. Thank you in advance.

Answer 1

Consider the issue of surjectivity. If you have $\lambda u - u' = f$ for $u \in D$ and $f \in X$ then you have $(e^{-\lambda t} u)' = -e^{-\lambda t} f$. Since $u(0) = 0$ this gives you the unique solution in $C^1([0, 1])$ of: $$u(t) = -e^{\lambda t} \int_0^t e^{-\lambda x} f(x) d x$$ Then note that say $f(x) = x$ is $C^1([0, 1])$, has $f(0) = 0$, and $u(0) = u'(0) = 0$. But $u(1) < 0$ for any $\lambda$.

To have a non-empty resolvent you could have:

• $D = \{u \in C^2([0, 1]) : u(0) = u'(0) = 0\}$
• $X = \{u \in C^1([0, 1]) : u(0) = 0\}$

which sidesteps this issue with $u(1)$, which will necessitate a "zero-average condition" of sorts.