K-Frame of multipliers in Hilbert pro-C*-module

‎For the study of atomic systems, first introduced by Feichtinger et al. Gavruta presented K-frames on Hilbert spaces. K-frames are a kind of frames in sense that the lower frame bound only holds for the elements in the range of the K, where K is a bounded linear operator in Hilbert space. C*-algebra whose topology is induced by a family of continuous C*-seminorms instead of a C*-norm is called pro-C*-algebra. Hilbert pro-C*-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a pro-C*-algebra rather than in the field of complex numbers. In this paper, the sequences whose elements are adjointable operators from pro-C*-algebra into Hilbert pro-C*-module is called the sequence of multipliers. We introduce the concept atomic systems and K-frame of multipliers in Hilbert pro-C*-modules and for more information, we give an example of K-frames. We obtain a condition that sequences of multipliers is frame, also we investigate the relationship between atomic systems and K-frames with each other and the frame of multipliers. If K is a bounded operator with certain conditions then every K-frame of multipliers is a frame of multipliers in Hilbert pro-C*-module. Also, we investigate some of the properties of these concepts, such as the combination of operators with K-frames in Hilbert pro-C*-module.