Generalized cline’s formula for g-Drazin inverse in a ring

In this paper, we give a generalized Cline?s formula for the generalized Drazin inverse. Let R be a ring, and let a, b, c, d ? R satisfying (ac)2 = (db)(ac), (db)2 = (ac)(db), b(ac)a = b(db)a, c(ac)d = c(db)d. Then ac ? Rd if and only if bd ? Rd. In this case, (bd)d = b((ac)d)2d: We also present generalized Cline?s formulas for Drazin and group inverses. Some weaker conditions in a Banach algebra are also investigated. These extend the main results of Cline?s formula on g-Drazin inverse of Liao, Chen and Cui (Bull. Malays. Math. Soc., 37(2014), 37-42), Lian and Zeng (Turk. J. Math., 40(2016), 161-165) and Miller and Zguitti (Rend. Circ. Mat. Palermo, II. Ser., 67(2018), 105-114). As an application, new common spectral property of bounded linear operators over Banach spaces is obtained.

Extensions of Cline’s formula for some new generalized inverses

Let a, b, c, d be elements in a unital associative ring R. In this note, we generalize Cline?s formula for some new generalized inverses such as strong Drazin inverse, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse to the case when acd = dbd and dba = aca. As a particular case, some recent results are recovered.

Generalized Cline’s formula and common spectral property

Let [Formula: see text] be an associative ring with an identity and suppose that [Formula: see text] satisfy [Formula: see text] If [Formula: see text] has generalized Drazin (respectively, p-Drazin, Drazin) inverse, we prove that [Formula: see text] has generalized Drazin (respectively, p-Drazin, Drazin) inverse. In particular, as applications, we obtain new common spectral property of bounded linear operators over Banach spaces.

Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

On gs-Drazin inverses in a ring

An element [Formula: see text] in a ring [Formula: see text] has a gs-Drazin inverse if there exists [Formula: see text] such that [Formula: see text]. In this paper, we extend Cline’s formula and Jacobson’s Lemma for gs-Drazin inverses. Various additive properties of gs-Drazin inverses are thereby obtained.

Cline’s formula for g-Drazin inverses in a ring

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

Formulating the Building Blocks for National Cyberpower

Cyber threats pose a growing risk to national security for all nations; cyberpower is consequently becoming an increasingly prominent driver in the attainment of national security for any state. This paper investigates the national cyberpower environment by analysing the elements of cyberspace as part of national security. David Jablonsky (1997) distinguishes between natural and social determinants of power in his discussion of national power. Also, Jablonsky refers to Ray Cline's formula (Cline, 1993) to determine a rough estimate of “perceived” national power by focusing primarily on a state's capacity to wage war. In this paper, the formula for Perceived Power (PP) will be adapted for use in cyberspace to create a similar formula for Perceived Cyberpower (PCP) that focuses primarily on a state's capacity for cyberwarfare. Military cyberpower is one of the critical elements of cyberpower. The paper also discusses how to operationalise military cyberpower.