Let $X$ be a compact plane set. $A(X)$ denotes the
uniform algebra of all continuous complex-valued functions on $X$
which are analytic on int$X$. For $0<\alpha\leq 1$, Lipschitz
algebra of order $\alpha$, $Lip(X,\alpha)$ is the algebra of all
complex-valued functions $f$ on $X$ for which
$p_\alpha(f)=\sup\{\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}:x,y\in X,
x\neq y\}<\infty.$
Let $Lip_A(X,\alpha)=A(X)\bigcap Lip(X,\alpha)$, and
$Lip^{n}(X,\alpha)$ be the algebra of complex-valued functions
on $X$ whose derivatives up to order $n$ are in $\Lip(X,\alpha)$.
$Lip_A(X,\alpha)$ under the norm $\|f\|=\|f\|_X+p_\alpha(f)$, and
$Lip^n(X,\alpha)$ for a certain plane set $X$ under the norm
$\|f\|=\sum_{k=0}^{n}\frac{\|f^{(k)}\|_X+p_{\alpha}(f^{(k)})}{k!}$
are natural Banach function algebras, where $\|f\|_X = \sup_{x\in
X } |f(x)|$.
In this note we study endomorphisms of algebras $Lip_A(X,\alpha)$
and $Lip^n(X,\alpha)$ and investigate necessary and sufficient
conditions for which these endomorphisms to be compact. Finally,
we determine the spectra of compact endomorphisms of these
algebras.