The index of some perturbation about elliptic operator with Robin boundary condition

Question

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all continuous functions on $[0, 1]$ with maximal norm. For $u\in C^{2}(\bar{I})$, define two operators $$ Au = \frac{\partial^{2}}{\partial x^{2}}u + \frac{\partial}{\partial x}u \quad (1) $$ And $$ Bu= \frac{\partial^{2}}{\partial x^{2}}u \quad (2)$$ Let $L$ be a closed operator from $C^{2}(\bar{I})$ into $C(\bar{I})$ such that $$ Lu(x) = f(x)\\ \frac{\partial u}{\partial x}(0)=0, \quad u(1)=0 \quad (3) $$ where $f\in C(\bar{I})$. Define $L_{0}$ from $C^{2}(\bar{I})$ into $C(\bar{I})$ as $$ L_{0}u=( \frac{\partial u}{\partial x}(0), u(1))$$ From (3) we can view $ (L, L_{0})$ as a linear map from Banach space $C^{2}(\bar{I})$ into Banach space $C(\bar{I})\bigoplus R \bigoplus R$ with usual norm. According to classic elliptic theory, when $L=A$, we can claim that $(A, L_{0})$ is an algebraic and topological isomorphic from $C^{2}(\bar{I})$ into Banach space $C(\bar{I})\bigoplus R \bigoplus R$. Therefor $$ index(A, L_{0})=0 $$ From (2), we can view $(B, 0)$ as a compact operator from $C^{2}(\bar{I})$ into Banach space $C(\bar{I})\bigoplus R \bigoplus R$. Therefore according to Theorem 5.26 in Perturbation Theory for Linear Operators (Classics in Mathematics, springer 1995) by Tosio Kato, we have $$ index (A-B, L_{0}) = index(A, L_{0}) = 0 $$ There is a contraction since $A-B$ is $\frac{\partial u}{\partial x}$ which says that $$ index(A-B, L_{0})=-1 $$ Where am I wrong? Any suggestions are very appreciated.

Answer 1

Your operator $B$ is not compact.