Operator Decompositions and Weighted Frame Inequalities for Total Sequences in Hilbert Spaces

We study total sequences in Hilbert spaces and their relation to frame inequalities. Building on the recent theorem of Ozawa, which established that every total sequence can be rescaled to satisfy the lower frame inequality, we develop an operator--decomposition framework that both clarifies the proof strategy and extends the result. Our approach constructs a tail--adapted orthonormal system and a sequence of tail--supported approximants, leading to a canonical rank--one resolution of the identity in the strong operator topology. From this decomposition we derive a general weighted lower frame inequality: for any sequence of positive weights, one obtains a rescaling of the total sequence that ensures frame--type stability. In addition, we provide explicit formulas for the frame coefficients, expressed in terms of perturbation bounds and tail expansions, which render the construction constructive and quantitative. Taken together, these results reorganize Ozawa's argument into a transparent operator framework, extend it to general weighted inequalities, and provide explicit coefficient formulas. The main conclusion is that total sequences are not only rescalable into frames, but their stability can be quantified and localized along the tails of the sequence.