Abstract: An upper bound is obtained for the distance between two hyponormal operators in terms of the distance between their spectra. Moreov er, for a hyponormal operator 〖WTHX〗A〖WT〗, the condition inf〖DD( 〗〖〗λ∈C〖DD)〗〖JB(=〗〖WTHX〗A〖WTBX〗-λ〖WTHX〗I〖WT〗〖JB)=〗=〖JB(=〗〖WTHX 〗A〖WT〗〖JB)=〗 is equivalent to ∩〖DD(〗〖〗x∈σ(〖WTHX〗A〖WTBX〗)〖 DD)〗U(x,〖JB(=〗〖WTHX 〗A〖WT〗〖JB)=〗)={0}, where U(x,〖JB(=〗〖WTHX〗A〖WT〗〖JB)=〗)={z∈C; 〖JB(|〗z-x〖JB)|〗≤〖JB(=〗〖WTHX〗A〖WT〗〖JB)=〗}.

Key words: normal operator, hyponormal operator, spectra, spectral radius