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Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of positive numbers and $1\le p<\mathrm{\infty }$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}\stackrel{⌃}{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\left|\stackrel{⌃}{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\mathrm{\infty }$ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and ${\sum }_{n=1}^{\mathrm{\infty }}\frac{n^{qj}}{\beta {\left(n\right)}^q}=\mathrm{\infty }$ for some nonnegative integer $j$. We show that if $C_{\phi }$ is compact on $H^p\left(\beta \right)$, then the non-tangential limit of ${\phi }^{\left(j+1\right)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\phi }$ is Fredholm on $H_p\left(\beta \right)$, then $\phi$ must be an automorphism of the open unit disc.