Controlled $g$-atomic subspaces for operators in Hilbert spaces

Controlled $g$-atomic subspace for a bounded linear operator is being presented and a characterization has been given. We give an example of controlled $K$-$g$-fusion frame. We construct a new controlled $K$-$g$-fusion frame for the Hilbert space $H \oplus X$ using the controlled $K$-$g$-fusion frames of the Hilbert spaces $H$ and $X$. Several useful resolutions of the identity operator on a Hilbert space using the theory of controlled $g$-fusion frames have been discussed. We introduce the frame operator for a pair of controlled $g$-fusion Bessel sequences.