Pseudo Core Inverses in Algebra

• Pseudo core inverses are a class of generalized inverses defined for elements in rings with involution, Banach *-algebras, and matrix algebras that extend core, group, and Moore–Penrose inverses.
• They are characterized by polynomial relations like x·a^(m+1)=a^m, symmetry under involution, and unique stabilization properties that ensure consistent representations.
• Applications span operator theory, matrix decompositions, and spectral analysis, offering computational techniques and unified frameworks for studying noninvertible elements.

Pseudo core inverses are a class of generalized inverses defined for elements in rings with involution, semigroups, Banach *-algebras, and matrix algebras, extending and unifying various prior notions such as the core inverse, core-EP inverse, group inverse, and Moore–Penrose inverse. They are characterized by polynomial relations, symmetry with respect to the involution, and stabilization properties, making them essential for the algebraic and operator-theoretic paper of noninvertible elements, particularly in the context of perturbation, decompositions, and spectral theory.

1. Definitions and Foundational Equations

For a ring $R$ with involution $*$, an element $a \in R$ is said to possess a pseudo core inverse if there exists $x \in R$ and an integer $m \geq 1$ such that:

• $x a^{m+1} = a^m$
• $a x^2 = x$
• $(a x)^* = a x$

The element $x$ is unique (for a fixed $m$) and denoted $a^{\circ D}$, with the minimal such $m$ being the pseudo core index, $I(a)$ (Gao et al., 2016). Equivalently, in Banach *-algebras and matrix settings, similar equations and uniqueness properties hold (Chen et al., 2022, Wang et al., 2018, Ferreyra et al., 2023).

For complex matrices, the pseudo core inverse coincides with the core-EP inverse—a concept defined for index-$k$ matrices by Manjunatha Prasad and Mohana (2014)—and it generalizes the core inverse originally introduced for index-1 matrices by Baksalary and Trenkler (Gao et al., 2016, Xu et al., 2017, Ferreyra et al., 2023).

2. Characterizations, Generalizations, and Connections

Several equivalent characterizations and explicit formulas for pseudo core inverses tie them firmly to other generalized inverses:

• $a$ is pseudo core invertible iff $a$ is Drazin invertible (with index $m$) and $a^k$ admits a ${1,3}$-inverse for some $k > m$; then

$a^{\circ D} = a^D a^k (a^k)^{(1,3)}$

where $a^D$ is the Drazin inverse, and $(1,3)$ indicates an outer inverse satisfying $a^k x a^k = a^k$ plus $(a^k x)^* = a^k x$ (Gao et al., 2016, Wang et al., 2018).

• Core inverses, group inverses, Moore–Penrose inverses, and dual core inverses interact as follows:
  • The core inverse in $R$ is defined by $a x a = a$, $x R = a R$, $R x = R a^*$;
  • Every core invertible element is group invertible; the pseudo core inverse generalizes core inverses to arbitrary index (Rakić et al., 2014, Xu et al., 2015).
• Further extensions such as the $w$-core inverse and $(b,c)$-core-EP inverse in semigroups and rings relate to the pseudo core inverse by weighting or parameterizing the core-like properties, unifying them under a general framework (Zhu et al., 2022, Zhu et al., 19 Dec 2024).

In additive categories with involution, the existence of pseudo core inverses is controlled by invertibility of morphism invariants involving kernels and cokernels, underlining deeper categorical generalizations (Li et al., 2018).

3. Additive, Absorption, and Block Properties

Pseudo core inverses admit rich additive properties:

• Additivity: If $a$ and $b$ in a Banach *-algebra or ring have pseudo core inverses and satisfy commutativity ($ab = ba$, $a^*b = ba^*$), then $a+b$ is pseudo core invertible under specified annihilation and spectral idempotent relations. Explicit formulas involve the pseudo core inverse of perturbations such as $f = a+b$, for example (Chen et al., 2022, Zhou et al., 6 Aug 2025):

$f^{\circ D} = (1 + a^{\circ D} b)^{-1} a^{\circ D}$

with $$a^{\circ D}(a+f)f^{\circ D} = a^{\circ D} + f^{\circ D}$$ if and only if $1 + a^{\circ D} b$ is invertible in $R$, mirroring known absorption laws for Drazin and Moore–Penrose inverses.

• Block Matrices: Sufficient structural and commutation conditions ensure block matrix operators ($M = [A\quad B; C\quad D]$) possess pseudo core inverses if $A$ and $D$ are pseudo core invertible and certain nilpotency and star relations hold (Chen et al., 2022, Bisht et al., 2016). Extensions to principal pivot transforms and Schur complements allow formulaic computation of the generalized inverse in block-algebraic settings.

4. Matrix Decompositions, Explicit Representations, and Computational Techniques

Pseudo core inverses admit effective computational methods in matrix algebra:

• Hartwig–Spindelböck Decomposition: For $A \in \mathbb{C}^{n \times n}$, decompose $$A = U \begin{bmatrix} D & K \ 0 & L \end{bmatrix} U^*$$, $U$ unitary. Then

$$A^{\circ D} = U [\text{explicit blocks involving } D,K,L] U^*$$

gives an explicit form for the pseudo core inverse (Gao et al., 2016).

• Core-EP Decomposition: For index-$k$ matrices,

$$A = U \begin{bmatrix} T & S \ 0 & N \end{bmatrix} U^*$$

($T$ nonsingular, $N$ nilpotent of index $k$), and the $m$-weak core inverse (generalizing pseudo core) is:

$A^{O_m} = (A^D)^{m+1} A^m P_{A^m}$

where $P_{A^m}$ is the orthogonal projector onto $R(A^m)$ (Ferreyra et al., 2023).

• Pseudo Principal Pivot Transform: Extending the group inverse setting, the block transform with pseudo core inverses yields formulas of the form

$$P_{\text{core}} = \begin{bmatrix} A^{\circ D} & -A^{\circ D}B \ C A^{\circ D} & D - C A^{\circ D} B \end{bmatrix}$$

with appropriate hypotheses (Bisht et al., 2016).

5. Applications, Structural Properties, and Algebraic Significance

Pseudo core inverses furnish a unified and extensible approach for the theory and computation of generalized inverses in diverse algebraic frameworks:

• For Banach *-algebras, pseudo core invertibility supports structural results for block operators and noncommutative analysis (Chen et al., 2022).
• In semigroups and monoids, pseudo core inverses (via the $(b,c)$-core-EP inverse) are connected to Green's relations and can be employed to solve constrained matrix equations, optimize approximations in the Euclidean norm, and establish rank criteria for invertibility (Zhu et al., 19 Dec 2024).
• In the context of tropical/supertropical algebra, the stabilization and closure properties of pseudo-inverses (and by analogy pseudo core inverses) allow spectral analysis where classical invertibility fails (Niv, 2013).

They preserve range and spectral features (range $R(a^{\circ D}) = R(a^k)$, spectral idempotency, and core partial orderings), maintain symmetry under the involution, and relate closely to the Drazin and Moore–Penrose inverses (e.g. EP elements occur when pseudo core and Moore–Penrose inverses coincide) (Wang et al., 2018, Gao et al., 2016).

6. Extensions: Weighted, One-Sided, and Parametrized Variants

Weighted core inverse ($w$-core inverse) and $(b,c)$-core-EP inverse generalize pseudo core invertibility by introducing parameters:

• $w$-core inverse: $a w x^2 = x$, $x a w a = a$, $(a w x)^* = a w x$
  • $a$ is pseudo core invertible iff $a^n$ is $a$-core invertible for some $n$ (Zhu et al., 2022).
• One-sided (right) pseudo core inverses admit analogous conditions with symmetry and polynomial relations imposed on one side (Wang et al., 2018).

These frameworks encapsulate the pseudo core inverse as a special case, provide explicit criteria in terms of invertibility along an element, and offer new computational avenues and relationships with other generalized inverses.

7. Summary and Outlook

Pseudo core inverses serve as a versatile bridge among generalized inverses, stabilizing range and spectral data, admitting rich additive and absorption laws, and providing computationally tractable matrix formulas. They subsume core, core-EP, $w$-core, $(b,c)$-core, and Drazin inverses within a single framework, facilitating analysis across algebraic, operator, and semigroup contexts. Recent advances detail their block, additive, and weighted versions, clarify connections via Green's relations, and extend their algebraic and operator-theoretic utility to Banach algebras, matrix theory, and tropical settings (Gao et al., 2016, Chen et al., 2022, Ferreyra et al., 2023, Zhu et al., 19 Dec 2024, Zhou et al., 6 Aug 2025).