A note on the Hyers-Ulam stability constants of closed linear operators

This paper concerns the properties of the Hyers--Ulam stability constant of closed linear operators. Using the Moore-Penrose inverse, we prove that the mapping $\overline{T}\rightarrow K_{\overline{T}}$ is lower semi-continuous and give some sufficient and necessary conditions for $\overline{T}\rightarrow K_{\overline{T}}$ to be continuous or locally bounded.} \par\noindent{\bf{\it Keywords}}: the constant of the Hyers--Ulam stability; Moore--Penrose inverse; lower semi-continuity; continuity; local boundedness.