LAUSR

Let $$\lambda _1, \lambda _2 \ldots \lambda _{n-1}$$ λ 1 , λ 2 … λ n - 1 are non-zero complex numbers, $$\lambda _n$$ λ n a complex number, n a positive integer, $$D^n$$ D n the n th order differential operator and $$T^n_{\lambda _1, \lambda _2 \ldots \lambda _{n}} = D^n+{\lambda _1}D^{n-1}+\cdots +{\lambda _{n-1}}D+\lambda _n I$$ T λ 1 , λ 2 … λ n n = D n + λ 1 D n - 1 + ⋯ + λ n - 1 D + λ n I , the linear combination of differential operators on the weighted Hardy space $$H^2(\beta ).$$ H 2 ( β ) . In this paper, we prove that, $$T^n_{\lambda _1, \lambda _2 \ldots \lambda _{n}}$$ T λ 1 , λ 2 … λ n n is bounded on $$H^2(\beta )$$ H 2 ( β ) if and only if D is bounded on $$H^2(\beta ).$$ H 2 ( β ) . Moreover, if $$\lambda _n = 0,$$ λ n = 0 , then $$T^n_{\lambda _1, \lambda _2 \ldots \lambda _{n}}$$ T λ 1 , λ 2 … λ n n is compact on $$H^2(\beta )$$ H 2 ( β ) if and only if D is compact on $$H^2(\beta ).$$ H 2 ( β ) . We also prove that $$D^j$$ D j has the Hyers–Ulam stability property on $$H^2(\beta )$$ H 2 ( β ) for each j . Furthermore, we prove that $$T^2_{0, 1}$$ T 0 , 1 2 fails to satisfy the Hyers–Ulam stability property on $$H^2(\beta )$$ H 2 ( β ) .