Abstract:

Let a: X →  Y, w:  Y →  X be morphisms in an additive category, k₁: K₁ →  X  be a kernel of (aw) ⁱ, k₂: K₂ → Y  be a kernel of (wa) ⁱ. Then the following propositions are equivalent: (1) a has a  w -weighted Drazin inverse a_d,w in £ ; (2)  ₁: X → L₁ is cokernel of (aw) ⁱ, k_{1 1} and (aw) ⁱ⁺¹+ ₁(k_{1 1}) ^-1k₁ are invertible; (3)  ₂ : Y →  L₂ is cokernel of (wa) ^j, k_{2  2} and (wa) ^j+1+ ₂(k_{2  2})^-1k₂ are invertible. And the Core-Nilpotent decomposition of w-weighted Drazin inverse of morphisms in the exact additive category £ with {1}-inverse is studied, the existence for the Core-Nilpotent decomposition of  w - weighted Drazin inverse of morphisms is proved. The extension of Drazin inverse of morphisms with kernels and its Core-Nilpotent decomposition are introduced and representations for its  w - weighted Drazin inverse and Core-Nilpotent decomposition are derived.

Key words: Exact additive category,  w-weighted Drazin inverse, Core-nilpotent decomposition