On frames for countably generated Hilbert $C^*$-modules

Abstract:

Let $V$ be a countably generated Hilbert $C^*$-module over a $C^*$-algebra $A.$ We prove that a sequence $\{f_i:i\in I\}\subseteq V$ is a standard frame for $V$ if and only if the sum $\sum _{i\in I}\langle x,f_i\rangle \langle f_i,x\rangle$ converges in norm for every $x\in V$ and if there are constants $C,D>0$ such that $C\Vert x\Vert ^2\le \Vert \sum _{i\in I}\langle x,f_i\rangle \langle f_i,x\rangle \Vert \le D\Vert x\Vert ^2$ for every $x\in V.$ We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert $C^*$-modules over the $C^*$-algebra of all compact operators on some Hilbert space is discussed.