具有核的态射的 w -加权Drazin逆

该文中, a: X→Y, w: Y→ X为加法范畴 £ 中的态射, k₁: K ₁→X是(aw)ⁱ 的核, k₂: K₂ →Y是(wa)^j 的核. 那么下列命题等价: (1) a 在 £ 中有w -加权Drazin逆a _d,w; (2) ₁:X→ L₁是(aw)ⁱ 的上核,k_{1 1}(aw)ⁱ⁺¹}+ ₁(k_{1 1})^-1k₁是可逆的; (3) ₂: Y→ L₂是(wa)^j 的上核, k_{2 2}和(wa)^j+1+ ₂(k_{2 2})^-1k₂是可逆的. 作者又研究了具有{1} -逆的正合加法范畴中态射的w -加权Drazin逆的柱心幂零分解, 证明了其存在性. 作者把具有核的态射的Drazin逆及其柱心幂零分解推广到具有核的态射的w -加权 Drazin逆及其柱心幂零分解, 并给出了表达式.

The w-weighted Drazin Inverse of Morphisms with Kernels

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.