Scalability of Generalized Frames for Operators

In this paper, the Parseval $K$-$g$-frames are constructed from a given $K$-$g$-frame by scaling the elements of the $K$-$g$-frame with the help of diagonal operators, and these frames are named scalable $K$-$g$-frames. Also, we prove some properties of scalable $K$-$g$-frames and construct new scalable $K$-$g$-frames from a given $K$-$g$-frame. The necessary and sufficient conditions for a $K$-$g$-frame to be scalable are given. Further, equivalent conditions for the scalability of $K$-$g$-frames and the $K$-frames induced by $K$-$g$-frames are obtained. Finally, it is shown that the direct sum of two scalable $K$-$g$-frames is again a scalable $K$-$g$-frame for some suitable bounded linear operator $K$.