Multipliers of Continuous K-Frames

We study multipliers Mm,f,g of continuous K-frames over a σ-finite measure space. We give criteria for invertibility on Ran(K) governed by ∥m−1∥∞, and show that previously published criteria stated in terms of ∥m∥∞ have hypotheses that no admissible data can satisfy, so that they are vacuous rather than false. A Douglas-type characterization replaces the known one: Mm,f,g is invertible on Ran(K) exactly when Ran(K)⊆Ran(πRan(K)Mm¯,g,f). We then characterize the existence of continuous K-right and K-left inverses by range inclusions, identify those of minimal norm, and represent one-sided inverses as multipliers of continuous K-duals; a two-dimensional example shows that a representation formula for K-left inverses previously stated in the discrete setting is not correct as stated. Finally, we prove that pointwise biorthogonality is ineffective when μ has no atoms, and replace it by a μ-weighted biorthogonality along a partition of Ω into blocks, which we then extend to families that are not constant on the blocks.