arXiv subjects
Publications and source records attributed to Enrico Boasso.
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.
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.
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.
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.
Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $ρ\colon L\to L(X)$, a representation of $L$ in $X$ such that $ρ(l)\in K(X)$ for all $l\in L$, the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $ρ$ are computed.
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.
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 $τ_{ST}\in L(L(Y,X))$, the elementary operator defined by $S$ and $T$, will be fully characterized.
In this work characterizations of Fredholm pairs and chains of Hilbert space operators are given. Following a well-known idea of several variable operator theory in Hilbert spaces, the aforementioned objects are characterized in terms of Fredholm linear and bounded maps. Furthermore, as an application of the main results of this work, direct proofs of the stability properties of Fredholm pairs and chains in Hilbert spaces are obtained.
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.
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.
A Banach space operator $T\in B(X)$ is left polaroid if for each $λ\in\hbox{iso}σ_a(T)$ there is an integer $d(λ)$ such that asc $(T-λ)=d(λ)<\infty$ and $(T-λ)^{d(λ)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-λ)<\infty$, $(T-λ)X$ is closed and $\dim(T-λ)^{-1}(0)<\infty$ at each $λ\in\hbox{iso }σ_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $τ_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$ if and only if $0\not\in\hbox{iso}σ_a(A\otimes B)$; a similar result holds for $τ_{AB}$ for finitely left polaroid $A$ and $B^*$.
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 $B$, $τ_{AB}=L_AR_B\in B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $A\otimes B$ and to $τ_{AB}$ will be given.
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.
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.
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.
The main objective of the present article is to characterize regular Fredholm pairs and chains in terms of Fredholm operators.
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.
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.