LAUSR

The purpose of the paper is to analyze g -frames of the form $$\{\varphi T^{i} \in B(\mathcal {H},\mathcal {K})\}_{i=0}^\infty $$ { φ T i ∈ B ( H , K ) } i = 0 ∞ , where $$T\in B(\mathcal {H})$$ T ∈ B ( H ) and $$\varphi \in B(\mathcal {H},\mathcal {K})$$ φ ∈ B ( H , K ) , and discuss the properties of the operator T . We consider stability of g -Riesz sequences of the form $$\{\varphi T^{i} \in B(\mathcal {H},\mathcal {K})\}_{i=0}^\infty $$ { φ T i ∈ B ( H , K ) } i = 0 ∞ . Finally, a weighted representation of a g frame is introduced and some of its properties are presented. We provide a sufficient condition for a given g -frame $$\Lambda =\{\Lambda _{i}\in {B(\mathcal {H},\mathcal {K})}\}_{i=1}^\infty $$ Λ = { Λ i ∈ B ( H , K ) } i = 1 ∞ to be represented by an operator $$T\in B(\mathcal {H})$$ T ∈ B ( H ) and a sequence $$\{a_i\}_{i=1}^\infty $$ { a i } i = 1 ∞ .