Generalized Weighted Core-EP Inverse

• The paper presents the generalized weighted core-EP inverse by modifying Hermitian conditions and index constraints with weights, yielding unique representations for square, rectangular, and Banach *-algebra cases.
• It details explicit formulas, projector representations, and decomposition theorems that link weighted core-EP inverses with weighted Moore–Penrose and Drazin constructions, aiding in structured matrix equation solutions.
• The methodology introduces parameterized extensions, such as the W‐weighted q‐BT and m‐weak core inverses, offering practical tools for system solving while outlining open issues in continuity and perturbation analysis.

Generalized weighted core-EP inverse denotes a family of weighted extensions of the core-EP inverse in which the EP-type Hermitian condition, the index constraint, and the range–nullspace prescription are modified by one or more weights. In the matrix setting this includes the $E$-weighted core-EP inverse $A^{,E}$, the $F$-weighted dual core-EP inverse $A^{F,}$, and the rectangular $W$-weighted core-EP inverse $A^{\oplus,W}$; in Banach $*$-algebras it appears as the $w$-weighted generalized core-EP inverse $a^{\tiny\textcircled{d},w}$ and, in a related formulation, as the associated inverse $a_w^{\circ e}$. Across these frameworks, the subject is organized around weighted Moore–Penrose and weighted Drazin constructions, explicit projector descriptions, decomposition theorems, and parameterized extensions such as the $A^{,E}$0-weighted $A^{,E}$1-BT inverse and the $A^{,E}$2-weighted $A^{,E}$3-weak core inverse (Behera et al., 2020, Gao et al., 2018, Chen et al., 10 Jul 2025, Chen et al., 14 Jul 2025, Ferreyra et al., 2024, Ferreyra et al., 2024).

1. Core definitions and ambient settings

The square-matrix formulation in “Further results on weighted core-EP inverse of matrices” introduces the generalized weighted Moore–Penrose inverse $A^{,E}$4 for $A^{,E}$5 as the matrix $A^{,E}$6 satisfying

$A^{,E}$7

For $A^{,E}$8 with $A^{,E}$9, the $F$0-weighted core-EP inverse $F$1 is defined by

$F$2

while the $F$3-weighted dual core-EP inverse $F$4 is defined by

$F$5

The same work also introduces the star $F$6-weighted core-EP matrix $F$7 and the star $F$8-weighted dual core-EP matrix $F$9 (Behera et al., 2020).

The rectangular-matrix theory uses a nonzero weight $A^{F,}$0 and the coupled square products $A^{F,}$1 and $A^{F,}$2. In “Representations and properties of the $A^{F,}$3-weighted core-EP inverse,” the $A^{F,}$4-weighted core-EP inverse $A^{F,}$5 is the unique $A^{F,}$6 satisfying

$A^{F,}$7

with $A^{F,}$8. Closely related formulations appear as the right and left weighted core-EP inverses $A^{F,}$9 and $W$0, defined respectively by $W$1 with $W$2, and by $W$3 with $W$4 (Gao et al., 2018, Kyrchei, 2020).

In Banach $W$5-algebras, “New properties of weighted generalized core-EP inverse in Banach algebras” defines the $W$6-weighted generalized core-EP inverse $W$7 as the unique $W$8 such that

$W$9

A related paper on generalized weighted EP elements denotes the associated inverse by $A^{\oplus,W}$0 and characterizes it by

$A^{\oplus,W}$1

This suggests that Banach-algebra treatments split into two closely aligned, but notationally distinct, generalizations of weighted core-EP behavior (Chen et al., 10 Jul 2025, Chen et al., 14 Jul 2025).

Setting	Notation	Defining conditions
Square weighted core-EP	$A^{\oplus,W}$2	$A^{\oplus,W}$3
Square weighted dual core-EP	$A^{\oplus,W}$4	$A^{\oplus,W}$5
Rectangular weighted core-EP	$A^{\oplus,W}$6	$A^{\oplus,W}$7
Banach-algebra generalized form	$A^{\oplus,W}$8	$A^{\oplus,W}$9 asymptotic quasinilpotent defect
Related Banach-algebra form	$*$0	$*$1 two Hermitian identities, $*$2

2. Existence, uniqueness, and structural characterizations

Existence and uniqueness are handled differently across weighted frameworks. For the generalized weighted Moore–Penrose inverse $*$3, existence is not guaranteed in general; if $*$4 and $*$5 are positive definite, $*$6 exists and is unique. For the $*$7-weighted core-EP inverse, Theorem 3.4 gives the range characterization

$*$8

and the dual characterization for $*$9 is

$w$0

If $w$1 or $w$2 exists, it is unique (Behera et al., 2020).

The rectangular $w$3-weighted core-EP inverse is also unique, but its existence is built into the finite-dimensional theory through the defining system. One formulation states that $w$4 is the unique $w$5 satisfying the three equations above; another gives the equivalent range–nullspace identities

$w$6

The right and left weighted core-EP inverses $w$7 and $w$8 likewise exist and are unique for any $w$9 with $a^{\tiny\textcircled{d},w}$0 (Gao et al., 2018, Kyrchei, 2020).

The weighted generalized families are more delicate. In “Characterizations of Weighted Generalized Inverses,” the classes $a^{\tiny\textcircled{d},w}$1 and $a^{\tiny\textcircled{d},w}$2 enlarge the core-EP setting by imposing weighted Penrose-type conditions together with an index equation. The paper explicitly notes that if $a^{\tiny\textcircled{d},w}$3, these classes contain infinitely many elements. A persistent misconception is therefore that every “generalized weighted core-EP” object remains unique; the literature shows that uniqueness is a property of the weighted core-EP inverse itself, not of all generalized families built around it (Sitha et al., 2023).

Banach-algebra generalizations preserve uniqueness at the inverse level. If $a^{\tiny\textcircled{d},w}$4 exists, then $a^{\tiny\textcircled{d},w}$5 exists and equals it; both defining elements are unique. In the generalized weighted EP formulation, existence of $a^{\tiny\textcircled{d},w}$6 is equivalent to $a^{\tiny\textcircled{d},w}$7 together with $a^{\tiny\textcircled{d},w}$8 being $a^{\tiny\textcircled{d},w}$9-EP, and uniqueness follows from the representation via the weighted generalized Drazin inverse (Chen et al., 10 Jul 2025, Chen et al., 14 Jul 2025).

3. Explicit formulas, decompositions, and projector representations

A central feature of the subject is the availability of explicit constructions through other generalized inverses. For square matrices, if $a_w^{\circ e}$0 and $a_w^{\circ e}$1, then

$a_w^{\circ e}$2

and if $a_w^{\circ e}$3 exists, then

$a_w^{\circ e}$4

For positive definite $a_w^{\circ e}$5, the paper further gives

$a_w^{\circ e}$6

It also proves the power relations

$a_w^{\circ e}$7

and the iteration identities

$a_w^{\circ e}$8

These formulas tie weighted core-EP inverses to Drazin stabilization and to weighted Moore–Penrose inverses of powers (Behera et al., 2020).

Rectangular theory emphasizes formulas involving only Moore–Penrose inverses, full-rank decompositions, and unitary decompositions. For $a_w^{\circ e}$9,

$A^{,E}$00

The same work gives full-rank and QR-based representations, and recommends the $A^{,E}$01-based formula when $A^{,E}$02 and the $A^{,E}$03-based formula when $A^{,E}$04. The simultaneous weighted core-EP decomposition and related block upper-triangularizations of the pair $A^{,E}$05 produce canonical block forms for $A^{,E}$06, for the $A^{,E}$07-weighted $A^{,E}$08-BT inverse, and for the $A^{,E}$09-weighted $A^{,E}$10-weak core inverse (Gao et al., 2018, Ferreyra et al., 2024, Ferreyra et al., 2024).

Projector descriptions are equally prominent. If $A^{,E}$11, then $A^{,E}$12 is a projector onto $A^{,E}$13 along $A^{,E}$14, while $A^{,E}$15 is a projector onto $A^{,E}$16 along $A^{,E}$17. In the rectangular setting,

$A^{,E}$18

is the orthogonal projector onto $A^{,E}$19, and $A^{,E}$20 is an oblique projector onto $A^{,E}$21 along a specified null space. This projector calculus is one of the main mechanisms by which weighted core-EP inverses are converted into subspace decompositions and outer inverses (Behera et al., 2020, Gao et al., 2018).

A further explicit direction is determinantal representation. Over the quaternion skew field, the right and left weighted core-EP inverses admit formulas in terms of noncommutative row and column determinants built from Gram-type matrices such as $A^{,E}$22 and $A^{,E}$23; over $A^{,E}$24, these specialize to ordinary determinant formulas. This line of work treats the weighted core-EP inverse together with weighted DMP, MPD, and CMP inverses (Kyrchei, 2020).

4. Generalized and parameterized extensions

Several recent constructions place the weighted core-EP inverse at the endpoint of a broader parameterized family. The $A^{,E}$25-weighted $A^{,E}$26-BT inverse is defined for $A^{,E}$27, $A^{,E}$28, $A^{,E}$29, and $A^{,E}$30 by

$A^{,E}$31

equivalently as the unique solution of

$A^{,E}$32

Its specializations are exact: $A^{,E}$33 gives $A^{,E}$34, $A^{,E}$35 gives the $A^{,E}$36-weighted BT inverse, and $A^{,E}$37 or $A^{,E}$38 gives the $A^{,E}$39-weighted core-EP inverse $A^{,E}$40. The theory requires only $A^{,E}$41, with no need for invertibility or rank constraints (Ferreyra et al., 2024).

The $A^{,E}$42-weighted $A^{,E}$43-weak core inverse provides another parameterization. It is defined by

$A^{,E}$44

where $A^{,E}$45 is the $A^{,E}$46-weighted $A^{,E}$47-weak group inverse. This inverse always exists and is unique. Its specializations are again exact: $A^{,E}$48 gives the rectangular extension of the WC inverse, while $A^{,E}$49 gives the $A^{,E}$50-weighted core-EP inverse $A^{,E}$51. The representation

$A^{,E}$52

shows that the entire family is organized around the weighted behavior of $A^{,E}$53 (Ferreyra et al., 2024).

A third extension scheme uses the generalized inverse of a square matrix $A^{,E}$54 with respect to another same-size matrix $A^{,E}$55,

$A^{,E}$56

Within this framework, choosing $A^{,E}$57 and $A^{,E}$58 yields

$A^{,E}$59

which is the paper’s notation for the $A^{,E}$60-weighted core-EP inverse, while choosing $A^{,E}$61 yields the $A^{,E}$62-weighted BT-inverse. This places weighted BT and weighted core-EP inverses inside a single “with respect to $A^{,E}$63” construction (Kara et al., 8 Jan 2025).

These parameterized extensions also delineate the limits of core-EP behavior. In the $A^{,E}$64-BT setting, the identities

$A^{,E}$65

fail in general for $A^{,E}$66, but hold at $A^{,E}$67, where $A^{,E}$68. This indicates that generalized weighted core-EP inverses should not be treated as merely “one more parameter value” inside every weighted inverse family; the stabilized index regime is structurally distinguished (Ferreyra et al., 2024).

5. Banach-algebra generalizations

The Banach $A^{,E}$69-algebra theory replaces matrix index conditions by quasinilpotent defects and weighted generalized Drazin structure. The main decomposition theorem for the $A^{,E}$70-weighted generalized core-EP inverse states that

$A^{,E}$71

and in that case

$A^{,E}$72

The stronger condition $A^{,E}$73 may also be imposed, but the paper shows it is dispensable. This decomposition is the Banach-algebra analogue of splitting an element into a weighted core-invertible part and a weighted quasinilpotent part (Chen et al., 10 Jul 2025).

A polar-like characterization is also available. For $A^{,E}$74,

$A^{,E}$75

if and only if $A^{,E}$76 and there exists a projection $A^{,E}$77 such that

$A^{,E}$78

The same paper proves the representation

$A^{,E}$79

and the alternative formula

$A^{,E}$80

This reduces computation of the generalized weighted core-EP inverse to the weighted generalized Drazin inverse plus a weighted core or weighted $A^{,E}$81-inverse of the Drazin part (Chen et al., 10 Jul 2025).

The related theory of generalized weighted EP elements uses the notation $A^{,E}$82 for the associated inverse. It establishes the equivalence

$A^{,E}$83

together with the representation

$A^{,E}$84

Its weighted core-EP decomposition takes the form

$A^{,E}$85

This suggests a close correspondence between generalized weighted core-EP inversion and weighted EP decomposition in Banach $A^{,E}$86-algebras, even when the terminology differs (Chen et al., 14 Jul 2025).

6. Applications, system solving, and open issues

Weighted core-EP inverses are repeatedly used to solve structured matrix equations. The star $A^{,E}$87-weighted core-EP matrix $A^{,E}$88 is the unique solution of

$A^{,E}$89

and the dual star matrix $A^{,E}$90 solves the corresponding dual system. The same theory establishes additivity: $A^{,E}$91 whenever $A^{,E}$92, $A^{,E}$93, and both weighted inverses exist; there is an analogous dual statement with $A^{,E}$94. These results connect weighted core-EP inversion to annihilating decompositions and to system-solving via star classes (Behera et al., 2020).

The rectangular parameterized families make the application side more explicit. For the $A^{,E}$95-weighted $A^{,E}$96-weak core inverse, the paper gives general and unique subspace solutions of linear systems such as

$A^{,E}$97

with solution

$A^{,E}$98

For the $A^{,E}$99-weighted $F$00-BT inverse, the range and null-space formulas

$F$01

show that the inverse acts as an outer inverse of $F$02 with prescribed geometric data (Ferreyra et al., 2024, Ferreyra et al., 2024).

Several limitations and open directions are explicit in the literature. One paper notes that continuity and perturbation bounds have been developed for core-EP inverses, but detailed perturbation results for weighted core-EP inverses are future work. The same work proposes iterative methods for estimating star weighted core-EP matrices, perturbation bounds, and reverse order laws as directions for further study. In the Banach-algebra setting, further continuity and perturbation bounds beyond additive $F$03-quasinilpotent stability, numerical schemes exploiting the representations via $F$04, and extension to pairs of weights $F$05 are identified as natural continuations (Behera et al., 2020, Chen et al., 10 Jul 2025).

A final misconception concerns the role of the weight. In the $F$06- and $F$07-weighted Moore–Penrose setting, positive definiteness of the weights guarantees existence and uniqueness; in the $F$08-weighted $F$09-BT theory, by contrast, the theory assumes only $F$10. This difference is not merely notational. It reflects distinct ambient geometries: one built from weighted Hermitian conditions on square matrices, the other from the coupled products $F$11 and $F$12 for rectangular pairs (Behera et al., 2020, Ferreyra et al., 2024).