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In this article, the properties of the (b, c)-inverse, the inverse along an element, the outer inverse with prescribed range and null space AT,S(2) and the Moore–Penrose inverse will be studied in ...

In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also researched in the context of rectangular matrices., Comment: 38 pages, original research article

[EN] Properties of the inverse along an element in rings with an involution, Banach algebras and C*-algebras will be studied unifying known expressions concerning generalized inverses., This work was supported by the Universidad Politecnica de Valencia [SP20120474].

[EN] The continuity of the core inverse and the dual core inverse is studied in the setting of -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 -algebras., The third author is supported by the Scientific Research Foundation for doctorate programme at Huaiyin Institute of Technology [grant number Z301B18534]. The third author is grateful to China Scholarship Council for helping him pursue his further study at Universitat Politècnica de València, Spain

Given Banach spaces $${\mathcal{X}}$$ and $${\mathcal{Y}}$$ and Banach space operators $${A \in L (\mathcal{X})}$$ and $${B \in L (\mathcal{Y})}$$ , let $${\rho \colon L (\mathcal{Y},\mathcal{X})\to L(\mathcal{Y},\mathcal{X})}$$ denote the generalized derivation defined by A and B, i.e., $${\rho (U)=AU-UB}$$ ( $${U\in L(\mathcal{Y},\mathcal{X})}$$ ). The main objective of this article was to study Weyl and Browder type theorems for $${\rho \in L (L(\mathcal{Y},\mathcal{X}))}$$ . To this end, however, first the isolated points of the spectrum and the Drazin spectrum of $${\rho\in L (L(\mathcal{Y},\mathcal{X}))}$$ need to be characterized. In addition, it will be also proved that if A and B are polaroid (respectively, isoloid), then ρ is polaroid (respectively isoloid).

Given Banach spaces $X$ and $Y$ and operators $A\in B(X)$ and $B\in B(Y)$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $A\otimes B\in B(X\overline{\otimes} Y)$ or to the elementary operator defined by $A$ and $B$, $\tau_{AB}=L_AR_B\in B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $A\otimes B$ and to $\tau_{AB}$ will be given., Comment: 14 pages, original research article

The transfer property for the generalized Browder’s theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the B -Weyl spectrum inclusion. In addition, the isolated points of these two classes of operators will be fully characterized.

Given two complex Banach spaces $X_1$ and $X_2$, a tensor product of $X_1$ and $X_2$, $X_1\tilde{\otimes}X_2$, in the sense of J. Eschmeier ([5]), and two finite tuples of commuting operators, $S=(S_1,\ldots ,S_n)$ and $T=(T_1,\ldots ,T_m)$, defined on $X_1$ and $X_2$ respectively, we consider the $(n+m)$-tuple of operators defined on $X_1\tilde{\otimes}X_2$, $(S\otimes I,I\otimes T)= (S_1\otimes I,\ldots ,S_n\otimes I,I\otimes T_1,\ldots ,I \otimes T_m)$, and we give a description of the semi-Browder joint spectra introduced by V. Kordula, V. M\"uller and V. Rako$\check{c}$evi$\acute{ c}$ in [7] and of the split semi-Browder joint spectra (see section 3), of the $(n+m)$-tuple $(S\otimes I ,I\otimes T)$, in terms of the corresponding joint spectra of $S$ and $T$. This result is in some sense a generalization of a formula obtained for other various Browder spectra in Hilbert spaces and for tensor products of operators and for tuples of the form $(S\otimes I ,I\otimes T)$. In addition, we also describe all the mentioned joint spectra for a tuple of left and right multiplications defined on an operator ideal between Banach spaces in the sense of [5]., Comment: 17 pages, original research article

Given two complex Hilbert spaces, $H_1$ and $H_2$, and two complex solvable finite dimensional Lie algebras of operators, $L_1$ and $L_2$, such that $L_i$ acts on $H_i$ (i= 1,2), the joint spectrum of the Lie algebra $L_1\times L_2$, which acts on $H_1\overline\otimes H_2$, is expressed by the cartesian product of $Sp(L_1,H_1)$ and $Sp(L_2,H_2)$., 7 pages, original research article

In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a fixed element will be fully described. Futhermore, commuting inverse along an element will be characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered., Comment: 17 pages, original research article

Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1\tilde{\otimes} X_2$ of $X_1$ and $X_2$ in the sense of [14], two complex solvable finite dimensional Lie algebras $L_1$ and $L_2$, and two representations $\rho_i\colon L_i\to {\rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1\times L_2$, and the tensor product representation of $L$, $\rho\colon L\to {\rm L}(X_1\tilde{\otimes}X_2)$, $\rho=\rho_1\otimes I +I\otimes \rho_2$. In this work we study the S{\l}odkowski and the split joint spectra of the representation $\rho$, and we describe them in terms of the corresponding joint spectra of $\rho_1$ and $\rho_2$. Moreover, we study the essential S{\l}odkowski and the essential split joint spectra of the representation $\rho$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $\rho_1$ and $\rho_2$. In addition, with similar arguments we describe all the above-mentioned joint spectra for the multiplication representation in an operator ideal between Banach spaces in the sense of [14]. Finally, we consider nilpotent systems of operators, in particular commutative, and we apply our descriptions to them., Comment: original research article, 46 pages

We consider the complex solvable non-commutative two dimensional Lie algebra $L$, $L=\oplus $, with Lie bracket $[x,y]=y$, as linear bounded operators acting on a complex Hilbert space $H$. Under the assumption $R(y)$ closed, we reduce the computation of the joint spectra $Sp(L,E)$, $\sigma_{\delta ,k}(L,E)$ and $\sigma_{\pi ,k}(L,E)$, $k= 0,1,2$, to the computation of the spectrum, the approximate point spectrum, and the approximate compression spectrum of a single operator. Besides, we also study the case $y^2=0$, and we apply our results to the case $H$ finite dimensional., Comment: 7 pages, original research article

The main objective of this article is to study several generalizations of the reverse order law for the Moore–Penrose inverse in ring with involution.

Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators., 15 pages; original research article

The main objective of this work is to study generalized Browder's and Weyl's theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $\tau_{A,B}=L_AR_B$., Comment: 16 pages, original research article

Given a Banach Algebra $A$ and $a\in A$, several relations among the Drazin spectrum of $a$ and the Drazin spectra of the multiplication operators $L_a$ and $R_a$ will be stated. The Banach space operator case will be also examined. Furthermore, a characterization of the Drazin spectrum will be considered., Comment: 16 pages; original research article

The main concern of this note is the Moore–Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C ∗ -algebra elements.

The main objective of this paper is to develop a notion of joint spectrum for complex solvable Lie algebras of operators acting on a Banach space, which generalizes the Taylor joint spectrum (T.J.S.) for several commuting operators., 7 pages, original research article

In this work, given a unital Banach algebra $\A$ and $a\in \A$ such that $a$ has a Moore-Penrose inverse $a^\dagger$, it will be characterized when $aa^\dagger-a^\dagger a$ is invertible. A particular subset of this class of objects will also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will be also considered., 14 pages, original research article

Given a (not necessarily continuous) homomorphism between Banach algebras $\T\colon\A\to\B$, an element $a\in\A$ will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to $\T$, if $\T(a)\in \B$ is Drazin invertible (respectively Koliha-Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered., Comment: 17 pages, original research article

Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements., 14 pages, original research article. arXiv admin note: text overlap with arXiv:1401.0469, arXiv:1306.4093

Given $L$ a solvable Lie Algebra of operators acting on a Banach space $E$, we study the action of the opposite algebra of $L$, $L'$, on $E^*$. Moreover, we extend S{\l}odkowski joint spectra, $\sigma_{\delta,k}$, $\sigma_{\pi,k}$, and we study its usual spectral properties., Comment: 10 pages, original research article

Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $\rho\colon L\to L(X)$, a representation of $L$ in $X$ such that $\rho (l)\in K(X)$ for all $l\in L$, the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $\rho$ are computed., Comment: 8 pages, original research article

Given unital Banach algebras $A$ and $B$ and elements $a\in A$ and $b\in B$, the Drazin spectrun of $a\otimes b\in A\overline{\otimes} B$ will be fully characterized, where $A\overline{\otimes} B$ is a Banach algebra that is the completion of $A\otimes B$ with respect to a uniform crossnorm. To this end, however, first the isolated points of the spectrum of $a\otimes b\in A\overline{\otimes} B$ need to be characterized. On the other hand, given Banach spaces $X$ and $Y$ and Banach space operators $S\in L(X)$ and $T\in L(Y)$, using similar arguments the Drazin spectrum of $\tau_{ST}\in L(L(Y,X))$, the elementary operator defined by $S$ and $T$, will be fully characterized., Comment: 12 pages, original research article

The weighted Moore-Penrose inverse will be introduced and studied in the context of Banach algebras. In addition, weighted EP Banach algebra elements will be characterized. The Banach space operator case will be also considered., 18 pages; original research article

For a complex nilpotent finite dimensional Lie algebra of matrices,and a Jordan-H\"older basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices., Comment: 13 pages, original research article

In this work it is proved that the Moore-Penrose inverse of the product of $n$-doubly commuting regular $C^*$-algebra elements obeys the so-called reverse order law. Conversely, conditions regarding the reverse order law of the Moore-Penrose inverse are given in order to characterize when $n$-regular elements doubly commute. Furthermore, applications of the main results of this article to normal $C^*$-algebra elements, to Hilbert space operators and to Calkin algebras will be considered., Comment: 12 pages, original research article

Several basic properties of the Drazin spectrum in Banach algebras will be studied. As an application, some results on meromorphic Banach space operators will be obtained., 8 pages, original research article

EP Banach space operators and EP Banach algebra elements are characterized using different kinds of factorizations. The results obtained generalize well-known characterizations of EP matrices, EP Hilbert space operators and EP $C^*$-algebra elements. Furthermore, new results that hold in these contexts are presented., 16 pages; original research article

In this work it is introduced the notion of regular Fredholm pair, i.e. a Fredholm pair whose operators are regular. The main properties of these objects are studied, and what is more, they are entirely classified. Furthermore, the index of a Fredholm pair turns out to be an extremely useful tool in the description of the aforementioned objects. Finally, regular Fredholm pairs are characterized in terms of regular Fredholm symmetrical pairs, exact chains of multiplication operators, and invertible Banach space operators., 23 pages, original research article

In this article, two results regarding the Moore-Penrose inverse in the frame of $C^*$-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is $C^*$-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact,these elements will be fully characterized both in the Hilbert space and in the $C^*$-algebra setting. Furthermore, it will be proved that an element is normal and Moore-Penrose hermitian if and only if it is a hermitian partial isometry., Comment: 14 pages, original research article

A Banach space operator $T\in B(X)$ is left polaroid if for each $\lambda\in\hbox{iso}\sigma_a(T)$ there is an integer $d(\lambda)$ such that asc $(T-\lambda)=d(\lambda), Comment: 10 pages, original research article

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered., 13 pages, original research article

In this work several results regarding the Cartan version of the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra will be studied., 11 pages, original research article

Given a complex nilpotent finite dimensional Lie algebra of linear transformations $L$, in a complex finite dimensional vector space $E$, we study the joint spectra $Sp(L,E)$, $\sigma_{\delta,k}(L,E)$ and $\sigma_{\pi,k}(L,E)$. We compute them and we prove that they all coincide with the set of weights of $L$ for $E$. We also give a new interpretation of some basic module operations of the Lie algebra $L$ in terms of the joint spectra., Comment: 10 pages, original research article