The　atomic　decomposition　of　signals　is　one　of　the　most　important　problems　in　the　frame　theory.　K-dual　frame　pairs　may　be　used　to　stably　reconstruct　elements　from　the　range　of　bounded　linear　operators　on　Hilbert　spaces.　The　purpose　of　this　paper　is　making　K-dual　frame　pairs　and　finding　common　K-dual　Bessel　sequence.　We　present　a　sufficient　condition　on　operators　on　(Formula　presented.)　which　takes　a　K-dual　frame　pairs　to　other　ones;　characterize　bounded　linear　operators　on　(Formula　presented.)　that　transform　K-dual　frame　pairs　to　other　ones;　prove　that　two　Bessel　sequences　can　always　be　extended　to　a　K-dual　frame　pair,　and　that　two　orthogonal　K-frames　have　a　common　K-dual　Bessel　sequence　under　certain　conditions;　and　obtain　a　sufficient　condition　which　the　K-duals　of　one　K-frame　is　contained　in　the　ones　of　another　K-frames.　Abundant　examples　are　also　provided　to　illustrate　the　generality　of　the　theory.　©　2021　Taylor　&　Francis　Group,　LLC.