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Register a new userFurther results on the $(b, c)$-inverse, the outer inverse $A^{(2)}_{T, S}$ and the Moore-Penrose inverse in the Banach context
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Authors
Enrico Boasso
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In this article properties of the $(b, c)$-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A^{(2)}_{T, S}$ and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^*$-algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the $(b, c)$-inverse and the outer inverse $A^{(2)}_{T, S}$ will be also characterized.
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In this article the $(b, c)$-inverse will be studied. Several equivalent conditions for the existence of the $(b,c)$-inverse in rings will be given. In particular, the conditions ensuring the existence of the $(b,c)$-inverse, of the annihilator $(b,c)$-inverse and of the hybrid $(b,c)$-inverse will be proved to be equivalent, provided $b$ and $c$ are regular elements in a unitary ring $R$. In addition, the set of all $(b,c)$-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the $(b,c)$-inverse and the Bott-Duffin inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally the continuity of the $(b,c)$-inverse will be characterized
In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on $H$, we give the expression of the Moore-Penrose inverse $bar{T}^dag$ and the upper bounds of $|bar{T}^dag|$ and $|bar{T}^dag -T^dag|$. These results obtained in this paper extend and improve many related results in this area.
A reflexive generalized inverse and the Moore-Penrose inverse are often confused in statistical literature but in fact they have completely different behaviour in case the population covariance matrix is not a multiple of identity. In this paper, we study the spectral properties of a reflexive generalized inverse and of the Moore-Penrose inverse of the sample covariance matrix. The obtained results are used to assess the difference in the asymptotic behaviour of their eigenvalues.
We present new additive results for the group inverse in a Banach algebra under certain perturbations. The upper bound of $|(a+b)^{#}-a^d|$ is thereby given. These extend the main results in [X. Liu, Y. Qin and H. Wei, Perturbation bound of the group inverse and the generalized Schur complement in Banach algebra, Abstr. Appl. Anal., 2012, 22 pages. DOI:10.1155/2012/629178].
The continuity of the core inverse and the dual core inverse is studied in the setting of C*-algebras. Later, this study is specialized to the case of bounded Hilbert space operators and to complex matrices. In addition, the differentiability of these generalized inverses is studied in the context of C*-algebras.
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