Searcharxiv⌕ Search

arXiv subjects

Enrico Boasso

Publications and source records attributed to Enrico Boasso.

At least 19 recordsLinked to original sources
arXiv2017

Further results on the $(b, c)$-inverse, the outer inverse $A^{(2)}_{T, S}$ and the Moore-Penrose inverse in the Banach context

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.

math.FA↗
arXiv2017

On one-sided (B,C)-inverses of arbitrary matrices

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.

math.RA↗
arXiv2016

Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

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 $ρ_i\colon L_i\to {\rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1\times L_2$, and the tensor product representation of $L$, $ρ\colon L\to {\rm L}(X_1\tilde{\otimes}X_2)$, $ρ=ρ_1\otimes I +I\otimes ρ_2$. In this work we study the Słodkowski and the split joint spectra of the representation $ρ$, and we describe them in terms of the corresponding joint spectra of $ρ_1$ and $ρ_2$. Moreover, we study the essential Słodkowski and the essential split joint spectra of the representation $ρ$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $ρ_1$ and $ρ_2$. 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.

math.FA↗
arXiv2016

The $(b, c)$-inverse in rings and in the Banach context

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

math.RA↗
arXiv2016

Tensor products and the semi-Browder joint spectra

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. Müller 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].

math.FA↗
arXiv2016

Joint spectrum for quasi-solvable Lie algebras of operators

Given a complex Banach space $X$ and a joint spectrum for complex solvable finite dimensional Lie algebras of operators defined on $X$, we extend this joint spectrum to quasi-solvable Lie algebras of operators, and we prove the main spectral properties of the extended joint spectrum. We also show that this construction is uniquely determined by the original joint spectrum.

math.FA↗
arXiv2016

On the joint spectra of the two dimensional Lie algebra of operators in Hilbert spaces

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 $Sp(L,E)$, $σ_{δ,k}(L,E)$ and $σ_{π,k}(L,E)$, $k= 0,1,2$, 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.

math.FA↗
arXiv2016

Joint spectra and nilpotent Lie algebras of linear transformations

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 $Sp(L,E)$, $σ_{δ,k}(L,E)$ and $σ_{π,k}(L,E)$. 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.

math.FA↗
arXiv2016

Tensor products and joint spectra for solvable Lie algebras of operators

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 $Sp(L_2,H_2)$.

math.FA↗
arXiv2015

The inverse along an element in rings

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.

math.RA↗
arXiv2015

A spectral theory for solvable Lie algebras of operators

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.

math.FA↗