Let　K　be　a　bounded　linear　operator　on　a　separable　Hilbert　space　H.　Gavruta　in　2012　proposed　the　notion　of　K-frame.　It　is　exactly　an　atomic　system　with　respect　to　K,　and　allows　a　stable　reconstruction　in　range(K).　This　paper　addresses　the　construction　of　K-frames.　We　characterize　bounded　linear　operators　on　l(2)　that　transform　a　pair　of　Bessel　sequences　into　a　K-frame;　present　a　sufficient　condition　to　obtain　K-frames　from　two　orthogonal　(disjoint)　K-frames.　Moreover,　when　dim(H)　＜　infinity,　we　investigate　Parseval　K-frames　and　K-frames　with　prescribed　norms.　Some　examples　are　also　provided.　(C)　2021　Elsevier　Inc.　All　rights　reserved.