m-Generalized Right Group Inverse

• The m-generalized right group inverse is a one-sided inverse in Banach *-algebras with proper involution, defined through intrinsic equations and asymptotic quasinilpotent conditions.
• It decomposes elements into a right-group invertible part and a quasinilpotent part using m-dependent orthogonality, thereby extending the m-weak group inverse framework.
• Its polar-like and idempotent characterizations, along with representations via the generalized right core inverse, provide practical tools for solving operator equations.

Searching arXiv for the cited paper and closely related work on generalized right group inverses. The $m$-generalized right group inverse is a one-sided generalized inverse defined for elements of a Banach $*$-algebra $\mathcal A$ with proper involution. It was introduced as a natural extension of both the $m$-weak group inverse and the generalized right group inverse, with the case $m=1$ recovering the ordinary generalized right group inverse. Its theory is organized around three equivalent viewpoints: an intrinsic system of equations, an $m$-generalized right group decomposition into a right-group-invertible part and a quasinilpotent part, and polar-like or idempotent characterizations; it also admits representations through generalized right core inverses (Chen et al., 12 Jul 2025, Chen et al., 16 Jul 2025).

1. Algebraic setting and precise definition

The theory is formulated in a Banach $*$-algebra $\mathcal A$ with proper involution. In the related generalized right group inverse framework, proper involution means

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$

and quasinilpotent elements are collected in

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

The notation $*$0 denotes generalized right Drazin invertible elements, and $*$1 denotes right group invertible elements (Chen et al., 12 Jul 2025, Chen et al., 16 Jul 2025).

An element $*$2 has an $*$3-generalized right group inverse if $*$4 and there exists $*$5 such that

$*$6

This $*$7 is called the $*$8-generalized right group inverse of $*$9, denoted by

$\mathcal A$0

The inverse is unique if it exists (Chen et al., 12 Jul 2025).

The definition is explicitly one-sided. The right-sided condition involving $\mathcal A$1 replaces the two-sided symmetry used in weak-group-type frameworks, while the asymptotic condition

$\mathcal A$2

plays the role of a quasinilpotent defect condition. This places the notion within the generalized right Drazin environment rather than within exact index-one invertibility alone (Chen et al., 12 Jul 2025).

2. Position in the hierarchy of generalized inverses

The $\mathcal A$3-generalized right group inverse was introduced as a natural extension of both the $\mathcal A$4-weak group inverse and the generalized right group inverse. In the same source, the $\mathcal A$5-weak group inverse is defined by

$\mathcal A$6

For matrices, this class coincides with the $\mathcal A$7-generalized group inverse. The new inverse is one-sided and therefore more flexible in the Banach algebra setting (Chen et al., 12 Jul 2025).

A central reduction theorem states

$\mathcal A$8

Thus the $\mathcal A$9-generalized right group inverse is controlled by the ordinary generalized right group inverse of $m$0. Taking $m$1 yields

$m$2

so the generalized right group inverse is the first member of the $m$3-family (Chen et al., 12 Jul 2025).

The ordinary generalized right group inverse itself is defined by the existence of $m$4 satisfying

$m$5

and is denoted $m$6. It extends both the generalized group inverse and the right group inverse in a proper Banach $m$7-algebra (Chen et al., 16 Jul 2025).

3. $m$8-generalized right group decomposition

The structural core of the theory is the $m$9-generalized right group decomposition. The main decomposition theorem states

$m=1$0

for some $m=1$1 such that

$m=1$2

In this situation,

$m=1$3

Accordingly, the inverse is extracted from the right-group-invertible component of a decomposition in which the residual part is quasinilpotent and orthogonal in the $m=1$4-dependent sense $m=1$5 (Chen et al., 12 Jul 2025).

This decomposition is the direct $m=1$6-dependent analogue of the generalized right group decomposition, where one has

$m=1$7

The difference is precisely the insertion of the factor $m=1$8 into the orthogonality condition, which controls how the higher-order parameter $m=1$9 interacts with the quasinilpotent component (Chen et al., 12 Jul 2025, Chen et al., 16 Jul 2025).

The decomposition theorem also yields basic identities for the inverse: $m$0 and

$m$1

These formulas show that the inverse behaves as a right-group-type outer inverse while retaining compatibility with higher powers of $m$2 (Chen et al., 12 Jul 2025).

4. Polar-like characterizations and representation through core-type inverses

A polar-like characterization is given in terms of an idempotent $m$3. One theorem states that

$m$4

if and only if there exists an idempotent $m$5 such that

$m$6

$m$7

$m$8

This says that the compression to the corner $m$9 is right invertible, while the $*$0-part is quasinilpotent and the left ideal generated by $*$1 is exactly $*$2 (Chen et al., 12 Jul 2025).

The same theorem has an equivalent $*$3-representation: $*$4 if and only if there exists $*$5 such that

$*$6

$*$7

Choosing $*$8 recovers the corresponding criterion for the ordinary generalized right group inverse (Chen et al., 12 Jul 2025).

A major representation theorem expresses the inverse through the generalized right core inverse. If $*$9 has generalized right core inverse $\mathcal A$0, defined by

$\mathcal A$1

then $\mathcal A$2 and

$\mathcal A$3

Further equivalent expressions include

$\mathcal A$4

These formulas show that the $\mathcal A$5-generalized right group inverse is computable from generalized right Drazin data and generalized right core data (Chen et al., 12 Jul 2025).

Earlier one-sided core-inverse theory provides the background for this representation. The right core inverse is characterized by

$\mathcal A$6

and the right pseudo core inverse is characterized by the condition that $\mathcal A$7 is right core invertible for some $\mathcal A$8. That earlier framework already emphasized power conditions, right-ideal equalities, and projection decompositions, all of which reappear in the $\mathcal A$9-generalized right group inverse setting (Wang et al., 2018).

5. Iteration in $x^*x=0 \implies x=0,\qquad x\in \mathcal A,$0, solvability, and explicit examples

The theory has a recursive aspect. A monotonicity theorem gives

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$1

with

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$2

Thus the sequence of $x^*x=0 \implies x=0,\qquad x\in \mathcal A,$3-generalized right group inverses is generated iteratively from the previous stage (Chen et al., 12 Jul 2025).

The paper also gives equivalent solvability criteria. One of them is

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$4

For regular elements $x^*x=0 \implies x=0,\qquad x\in \mathcal A,$5, another criterion states

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$6

with

$x^*x=0 \implies x=0,\qquad x\in \mathcal A,$7

These formulas connect the existence problem to right generalized Drazin invertibility and to recursive reduction by an inner inverse $x^*x=0 \implies x=0,\qquad x\in \mathcal A,$8 (Chen et al., 12 Jul 2025).

A concrete operator example is given on $x^*x=0 \implies x=0,\qquad x\in \mathcal A,$9. Define

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

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

Then

$$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$2

The paper notes that $$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$3 has generalized right core inverse, and hence it has an $$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$4-generalized right group inverse: $$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$5 acting by

$$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$6

The representation theorem is also used to obtain solutions of equations such as

$$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$7

This example and these formulas show that the theory applies to concrete bounded operators, not only to abstract algebraic elements (Chen et al., 12 Jul 2025).

6. Related block-matrix frameworks and terminological distinctions

The generalized right group inverse admits block-triangular formulas that clarify how one-sided inverse structure behaves under coupling. If

$$\mathcal A^{qnil}=\{x\in\mathcal A:\ \lim_{n\to\infty}\|x^n\|^{1/n}=0\}.$$8

and

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

then

$*$00

and

$*$01

A lower triangular analogue is also given. These formulas belong to the $*$02 theory, but by the equivalence

$*$03

they provide the immediate block-level context for higher-order variants (Chen et al., 16 Jul 2025, Chen et al., 12 Jul 2025).

For anti-triangular operator matrices, the terminology “$*$04-generalized right group inverse” is not used, but the same structural ingredients recur. The operator-matrix paper on

$*$05

and

$*$06

uses the spectral idempotent

$*$07

together with annihilation conditions such as

$*$08

to derive explicit generalized Drazin inverse and group inverse criteria. The paper explicitly notes that, while it does not use $*$09-generalized right group inverse language, its projection-based and index-controlled conditions are the right structural objects to compare with such inverses (Chen et al., 2023).

A second matrix-related comparison comes from the block formula for the group inverse of

$*$10

where, assuming $*$11 and $*$12 exist for

$*$13

one obtains

$*$14

under range inclusions such as

$*$15

That framework belongs to ordinary group inverse theory and pseudo principal pivot transforms, but it provides the block-matrix calculus out of which right-group-type decompositions are often built (Bisht et al., 2016).

The phrase “right generalized inverse” also appears in semigroup theory with a different meaning. A right generalized inverse semigroup is a regular semigroup whose idempotents form a right normal band, satisfying

$*$16

Its structure is governed by free left étale actions of inverse semigroups, and every right generalized inverse semigroup is isomorphic to a right Yamada semigroup (Kudryavtseva et al., 2012). This is a distinct notion from the $*$17-generalized right group inverse in Banach $*$18-algebras. A plausible implication is that the literature requires terminological care: identical adjectives can refer either to a one-sided inverse operation on algebra elements or to a structural class of semigroups.