Further results on the m-weak core inverse

This work establishes multiple novel representations for the m-weak core inverse, accompanied by proofs of their validity. Furthermore, we derive perturbation bounds and analyze continuity properties for this generalized inverse. By utilizing the m-weak core inverse, we characterize the unique solution to a constrained minimization problem in the Frobenius norm framework: min ǁ M^{{m+1}} X- M²ᵐ (Mᵐ) †B ǁF² , subject to the range constraint R(X) ⊆R( Mᵏ ), where m∈ N , B ∈ C ^n×q , M∈ C^n×n and ind(M)=k.