Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the unit interval. I need reference (or proof) for the following result:

\begin{align} ||R(\lambda,A_p)||_{\mathcal{L}(L_p)}\leq\frac{C}{dist(\lambda,\sigma(A_p))}, \ \lambda\in\rho(A_p). \end{align}

I am interested especially in the case when $\lambda$ lies between two consecutive eigenvalues of $A_p$. When $p=2$ the result follows with $C=1$ from the spectral theorem.