Weighted Generalized Drazin Inverse

• Weighted generalized Drazin inverse is defined in a Banach *-algebra using mixed products with a prescribed weight and quasinilpotent defect conditions.
• It underpins generalized weighted core-EP and EP inverses by facilitating a decomposition into regular and quasinilpotent components.
• Finite-dimensional realizations in matrices, tensors, and quaternions offer explicit formulas and computational methods via weighted Drazin-type calculus.

Searching arXiv for papers on weighted generalized Drazin inverse and closely related weighted inverse frameworks. Search query: all:"weighted generalized Drazin inverse" OR all:"weighted g-Drazin" OR ti:"weighted generalized Drazin inverse" The weighted generalized Drazin inverse is a weighted extension of the generalized Drazin inverse in a complex Banach *-algebra with identity. For a prescribed weight $w\in\mathcal A$, an element $a\in\mathcal A$ has weighted generalized Drazin inverse $a^{d,w}$ if there exists a unique $x\in\mathcal A$ such that

$$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$

Here $\mathcal A^{qnil}$ is the set of quasinilpotent elements. When $w=1$, this notion reduces to the generalized Drazin inverse in a Banach algebra. In current weighted inverse theory, $a^{d,w}$ serves as the principal Drazin-type object behind generalized weighted core-EP and generalized weighted EP constructions, while adjacent matrix, tensor, and quaternion literatures develop finite-dimensional weighted Drazin calculi that play an analogous structural role (Chen et al., 10 Jul 2025).

1. Definition and ambient structures

The Banach-algebraic setting uses two quasinilpotent classes. Besides

$$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$

one also considers the $w$-quasinilpotent set

$a\in\mathcal A$0

The weighted generalized Drazin inverse is defined directly in $a\in\mathcal A$1 through the mixed products $a\in\mathcal A$2 and $a\in\mathcal A$3, and its uniqueness is part of the definition of the class $a\in\mathcal A$4 of weighted g-Drazin invertible elements. This formulation is explicitly used in recent Banach *-algebra work on weighted core-EP and weighted EP structures (Chen et al., 10 Jul 2025).

A complementary viewpoint is furnished by the weighted Banach algebra $a\in\mathcal A$5, obtained from the same underlying vector space with multiplication

$a\in\mathcal A$6

and norm $a\in\mathcal A$7. In that framework, spectral data for $a\in\mathcal A$8 in $a\in\mathcal A$9 are controlled by the ordinary spectra of $a^{d,w}$0 and $a^{d,w}$1; specifically,

$a^{d,w}$2

and therefore

$a^{d,w}$3

This equivalence is the basic bridge between weighted and unweighted Drazin-type constructions in the weighted-algebra approach (Biswas et al., 4 Jun 2025).

The finite-dimensional matrix literature often uses the $a^{d,w}$4-weighted Drazin inverse rather than the Banach-algebraic weighted generalized Drazin inverse. For $a^{d,w}$5 and $a^{d,w}$6, the $a^{d,w}$7-weighted Drazin inverse is the unique $a^{d,w}$8 satisfying

$a^{d,w}$9

where $x\in\mathcal A$0. This is a weighted Drazin inverse with finite index, not the Banach-algebraic weighted generalized Drazin inverse defined by a quasinilpotent defect (Kyrchei, 2015).

2. Canonical decomposition and quasi-polar structure

A central structural result links weighted generalized Drazin theory to generalized weighted core-EP invertibility. If $x\in\mathcal A$1 has a generalized $x\in\mathcal A$2-weighted core-EP inverse, then $x\in\mathcal A$3 admits a decomposition

$x\in\mathcal A$4

where $x\in\mathcal A$5 is $x\in\mathcal A$6-weighted core invertible, $x\in\mathcal A$7, and $x\in\mathcal A$8; an equivalent stronger form also imposes $x\in\mathcal A$9. In this decomposition, the generalized weighted core-EP inverse is exactly the weighted core inverse of the regular part $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$0. This realizes the weighted generalized Drazin paradigm as a decomposition into a regular component and a $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$1-quasinilpotent component, directly paralleling ordinary generalized Drazin theory (Chen et al., 10 Jul 2025).

The same theory has a quasi-polar characterization. For $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$2 and $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$3, generalized weighted core-EP invertibility is equivalent to the conjunction of weighted generalized Drazin invertibility and the existence of a projection $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$4 such that

$$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$5

This identifies a weighted generalized Drazin part together with a projection separating a quasinilpotent weighted corner from an invertible perturbation of $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$6. The same paper also proves that the spectral idempotent

$$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$7

controls the generalized weighted core-EP projections through the equivalences

$$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$8

Under the condition $$awx=xwa,\qquad xwawx=x,\qquad a-awxwa\in\mathcal A^{qnil}.$$9, this yields explicit formulas for generalized weighted core-EP inverses of upper and lower triangular $\mathcal A^{qnil}$0 block matrices (Chen et al., 10 Jul 2025).

3. Relations with weighted core, core-EP, and EP inverses

The weighted generalized Drazin inverse is most fully developed as the organizing object behind weighted core-EP and weighted EP theories. The $\mathcal A^{qnil}$1-weighted core inverse of $\mathcal A^{qnil}$2 is the unique $\mathcal A^{qnil}$3 satisfying

$\mathcal A^{qnil}$4

and it exists exactly when $\mathcal A^{qnil}$5 and $\mathcal A^{qnil}$6. The generalized $\mathcal A^{qnil}$7-weighted core-EP inverse replaces the finite algebraic relation by the asymptotic condition

$\mathcal A^{qnil}$8

The decisive structural theorem states that

$\mathcal A^{qnil}$9

with explicit representation

$w=1$0

An equivalent formulation replaces weighted core invertibility of $w=1$1 by the existence of a weighted $w=1$2-inverse and gives

$w=1$3

Thus generalized weighted core-EP invertibility is strictly stronger than weighted g-Drazin invertibility, but it is characterized entirely in terms of the weighted generalized Drazin inverse and an additional core-type regularity on $w=1$4 (Chen et al., 10 Jul 2025).

A parallel development holds for generalized weighted EP elements. In that setting, a generalized $w=1$5-EP element $w=1$6 is characterized by the existence of $w=1$7 such that

$w=1$8

together with a quasinilpotent limit condition. The corresponding theorem states that

$w=1$9

In this case,

$a^{d,w}$0

The weighted generalized Drazin inverse is therefore the Drazin-type core of generalized weighted EP theory as well. The same paper further shows that weighted *-DMP elements are exactly those elements that are simultaneously generalized weighted EP and weighted Drazin invertible: $a^{d,w}$1 This separates the generalized, quasinilpotent level from the finite-index weighted Drazin level (Chen et al., 14 Jul 2025).

4. Weighted algebra formulations and higher Drazin-type variants

A significant generalization replaces the weighted generalized Drazin inverse by weighted generalized $a^{d,w}$2-strong Drazin inverses. In the weighted algebra $a^{d,w}$3, an element $a^{d,w}$4 is weighted generalized $a^{d,w}$5-strongly Drazin invertible if there exists $a^{d,w}$6 such that

$a^{d,w}$7

This is exactly generalized $a^{d,w}$8-strong Drazin invertibility in $a^{d,w}$9. The theory proves that

$$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$0

and the weighted inverse is given by

$$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$1

It also admits a purely quasinilpotent characterization: $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$2 The pseudo $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$3-strong version replaces quasinilpotence by membership in the radical hull $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$4, and similarly satisfies

$$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$5

These constructions were presented as weighted generalizations of generalized $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$6-strong and pseudo $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$7-strong Drazin inverses, extending the weighted g-Drazin perspective through the parameter $$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$8 (Biswas et al., 4 Jun 2025).

Within this framework, the weight is not merely auxiliary. The equivalence

$$\mathcal A^{qnil}=\left\{x\in\mathcal A:\lim_{n\to\infty}\|x^n\|^{1/n}=0\right\},$$9

allows weighted Drazin-type properties to be transported to ordinary elements $w$0 or $w$1, and all additive results established for generalized $w$2-strong and pseudo $w$3-strong Drazin inverses in the unweighted algebra carry over to the weighted setting by working in $w$4 (Biswas et al., 4 Jun 2025).

5. Matrix, tensor, and quaternion realizations

Finite-dimensional realizations of weighted Drazin-type inverses supply explicit formulas, canonical forms, and solution methods for restricted equations. For complex matrices $w$5 and $w$6, the $w$7-weighted Drazin inverse $w$8 admits limit representations

$w$9

and

$a\in\mathcal A$00

together with determinantal formulas for each entry and Cramer-type rules for restricted systems such as $a\in\mathcal A$01. These results belong to weighted Drazin theory rather than Banach-algebraic weighted generalized Drazin theory, but they provide an exact computational model for weighted singular inverses in finite dimensions (Kyrchei, 2015).

The tensor literature extends the same philosophy to even-order tensors under the Einstein product. For tensors $a\in\mathcal A$02 and $a\in\mathcal A$03, the $a\in\mathcal A$04-weighted Drazin inverse $a\in\mathcal A$05 is defined by weighted Drazin-type equations for $a\in\mathcal A$06 and $a\in\mathcal A$07, and the theory proves the explicit representation

$a\in\mathcal A$08

This yields tensor analogues of matrix weighted Drazin formulas and embeds the weighted inverse into the broader tensor generalized inverse framework (Behera et al., 2019).

Over the quaternion skew field, the $a\in\mathcal A$09-weighted Drazin inverse is defined for $a\in\mathcal A$10, $a\in\mathcal A$11 by the conditions

$a\in\mathcal A$12

and is related to ordinary Drazin inverses by

$a\in\mathcal A$13

Because quaternion multiplication is noncommutative, explicit formulas are expressed through row and column determinants of Hermitian quaternion matrices. This produces determinantal representations of $a\in\mathcal A$14 and Cramer-type formulas for the restricted equations $a\in\mathcal A$15, $a\in\mathcal A$16, and $a\in\mathcal A$17 (Kyrchei, 2015, Kyrchei, 2015).

6. Structural extensions, order laws, and scope

Several recent developments enlarge the weighted Drazin landscape without always using the Banach-algebraic weighted generalized Drazin definition. One line studies the $a\in\mathcal A$18-weighted $a\in\mathcal A$19-weak group inverse $a\in\mathcal A$20, defined by

$a\in\mathcal A$21

This inverse interpolates between the weighted weak group inverse ($a\in\mathcal A$22) and the $a\in\mathcal A$23-weighted Drazin inverse ($a\in\mathcal A$24, where $a\in\mathcal A$25). The theory gives projector identities, outer-inverse descriptions with prescribed range and null space, and canonical forms via the weighted core-EP decomposition (Gao et al., 2023).

A second line replaces weighted Drazin invertibility by minimal-rank weighted weak Drazin invertibility. In this framework, a minimal rank $a\in\mathcal A$26-weighted weak Drazin inverse satisfies a weak Drazin equation together with a rank-minimality condition, and it becomes the base object for $a\in\mathcal A$27-weighted weak MPD and DMP inverses. These constructions generalize the $a\in\mathcal A$28-weighted Drazin inverse, the $a\in\mathcal A$29-weighted core-EP inverse, and the $a\in\mathcal A$30-weighted $a\in\mathcal A$31-weak group inverse, and the theory develops projection characterizations, perturbation formulas, reverse and forward order laws, triple product laws, and explicit solution formulas for weighted matrix equations (Senapati et al., 30 Dec 2025).

The scope of the term itself is not uniform across the literature. In Banach *-algebras, the weighted generalized Drazin inverse is defined by the quasinilpotent defect condition $a\in\mathcal A$32. By contrast, substantial matrix, tensor, and quaternion literatures focus on finite-index $a\in\mathcal A$33-weighted Drazin inverses or weaker rank-minimal variants. A further boundary is visible in product-formula papers such as “Cline’s formula for G-Drazin inverse,” which do not define any weighted Drazin or weighted g-Drazin inverse; there the weighted setting appears only as a structural template obtainable by similarity, and the paper explicitly states that such weighted conclusions must be inferred rather than quoted directly (Chen et al., 2018). This suggests a bifurcated theory: a Banach-algebraic weighted generalized Drazin inverse centered on quasinilpotence and spectral decomposition, and a broader finite-dimensional weighted Drazin-type calculus centered on indices, explicit formulas, and structured matrix equations.