In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a is a Banach algebra has g-Drazin inverse if and only if there exists $x\in A$ such that $ax=xa, a-a^2x\in A^{qnil}$. we obtain the sufficient and necessary conditions for the existence of the g-Drain inverse for certain $2 \times 2$ anti-triangular matrices over a Banach algebra. These extend the results of Koliha (Glasgow Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra,27(1999), 3583-3592 and Zou et al. (Studia Scient. Math. Hungar., 54(2017,489-508).
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.
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 objective of this paper is to study the existence of the generalized Drazin inverse of the sum $a+b$ in a Banach algebra and present explicit expressions for the generalized Drazin inverse of this sum, under new conditions.
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.
In this paper, we introduce and study a new generalized inverse, called ag-Drazin inverses in a Banach algebra $\mathcal{A}$ with unit $1$. An element $a\in\mathcal{A}$ is ag-Drazin invertible if there exists $x\in\mathcal{A}$ such that $ax=xa, \, xax=x \ {\rm and} \ a-axa\in\mathcal{A}^{acc}$, where $\mathcal{A}^{acc}\triangleq\{a\in\mathcal{A}: a-λ1 \ {\rm is \ generalized \ Drazin\ invertible} \ {\rm for \ all} \ λ\in\mathbb{C}\backslash\{0\}\}.$ Using idempotent elements, we characterize this inverse and give some its representations. Also, we prove that $a\in\mathcal{A}$ is ag-Drazin inve…
Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$ where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.
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 $τ_{ST}\in L(L(Y,X))$,…
Let $R$ be an associative ring with an identity and suppose that $a,b,c,d \in R$ satisfy $bdb = bac,dbd = acd$. If $ac$ has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that $bd$ has generalized Drazin (respectively, pseudo Drazin, Drazin) inverse. In particular, as applications, we obtain new common spectral properties of bounded linear operators over Banach spaces.
In a Banach algebra, we introduce a new type of generalized inverse called g$π$-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we discuss the relationship between the g$π$-Hirano invertibility of $a$, $b$ and that of the sum $a+b$ under some weaker conditions. Finally, as applications to the previous additive results, some equivalent characterizations for the g$π$-Hirano invertibility of the anti-triangular matrix over Banach algebras are obtained.In particular, some results in this paper are different from the correspo…
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.
Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $λ\in \cit$ such that $T$ does not have the single-valued extension property at $λ$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.
A Hilbert space operator $A\in\B$ is left $(X,m)$-invertible by $B\in\B$ (resp., $B\in\B$ is an $(X,m)$-adjoint of $A\in\B$) for some operator $X\in\B$ if $\triangle_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{m-j}XA^{m-j}=0$ (resp., $δ_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{(m-j)}XA^j=0$). No Drazin invertible operator $A\in\B$, with Drazin inverse $A_d$, can be left $(I,m)$-invertible (equivalently, $m$-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin inverrtible operat…
Mosic and Djordjevic introduced the notation of the gDMP inverse for Hilbert space operators in [J. Spectr. Theory, 8(2):555-573, 2018] by considering generalized Drazin inverse with the Moore-Penrose inverse. This paper introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Hilbert space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed, along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit …
In this paper, we present new necessary and sufficient conditions under which the sum of two group invertible elements in a Banach algebra has group inverse. We then apply these results to block operator matrices over Banach spaces. The group inverses of certain operator block matrices are thereby obtained. Additionally, this paper extends the results obtained in \cite{L}.
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 invers…
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.
Properties of the inverse along an element in rings with an involution, Banach algebras and $C^*$-alegbras will be studied unifying known expressions concerning generalized inverses.
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 charact…
We study the regularity properties of the inverse of a bilipschitz mapping $f$ belonging $W^m X_{\text{loc}}$, where $X$ is an arbitrary Banach function space. Namely, we prove that the inverse mapping $f^{-1}$ is also in $W^m X_{\text{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in $W^m X_{\text{loc}}$ is closed with respect to composition and multiplication.
We prove that an interpolation pair of Banach lattices is uniquely determined by the collection of intermediate spaces with the property that these are interpolation spaces for positive operators. A correspondent result for exact interpolation is also presented.
In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.
In this paper we investigate Hirano invertibility of anti-triangular matrix over a Banach algebra. Let $a\in {\mathcal A}^H, b\in {\mathcal A}^{sD}.$ If $b^Da=0, bab^π=0,$ we prove that $\begin{pmatrix} a&1\\ b&0 \end{pmatrix}\in M_2(\mathcal A)^H.$ Moreover, we considered Hirano invertibility of anti-triangular matrices under commutative-like conditions. These provide new kind of operator matrices with tripotent and nilpotent decompositions.
Let $A$ be a Banach Algebra, we say that $a\in A$ has generalized Hirano inverse if there exists some $b \in A$ such that $b=b.a.b, a.b=b.a$ and $a-a^2.b$ is quasi nil potent, if and only if there exists some $p=p^3\in A$ such that $a-p$ is quasi nil potent.
In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator $\mathcal{A}=-\fracΔ{2}-x\nabla$ in the inverse Gaussian setting for Banach valued functions. We characterize the uniformly convex and smooth Banach spaces by using $L^p(\mathbb{R}^n,γ_{-1})$- properties of the $\mathcal{A}$-Littlewood-Paley functions. We also use Littlewood-Paley functions associated with $\mathcal{A}$ to characterize the Köthe function spaces with the UMD property.
In this paper we study higher order Riesz transforms associated with the inverse Gaussian measure given by $π^{n/2}e^{|x|^2}dx$ on $\mathbb{R}^n$. We establish $L^p(\mathbb{R}^n,e^{|x|^2}dx)$-boundedness properties and obtain representations as principal values singular integrals for the higher order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.
The application of generalized inverses is usually neglected in pure mathematical research. However, it is very effective for this paper. We expand the famous matrix rank theorem due to R. Penrose to operators between Banach paces. Therefore a modern perturbation analysis of generalized inverses is built. Hereby, we find and prove a complete rank theorem in advanced calculus. So a complete answer to the rank theorem problem presented by M. S. Berger is given. Applying the co-final set and the perturbation analysis of generalized inverses we prove the Frobenius theorem in Banach space, in the p…
Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on connection between the degree of smoothness of vector $x\in X$ with respect to operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for operator A, and the k-module of continuity are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtai…
A discrete group $\G$ is called rigidly symmetric if for every $C^*$-algebra $\A$ the projective tensor product $\ell^1(\G)\widehat\otimes\A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{α,ø}(\G;\A)$ is also a symmetric Banach $^*$-algebra, for every twisted action $(α,ø)$ of $\G$ in a $C^*$-algebra $\A$\,. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra …
Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semi-discrete inverse problem, in certain functional spaces. The choice of an appropriate space is crucial for solutions of these problems. Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems…
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.
Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $λ\rightarrow T-λS$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators a…
A surjective bounded homomorphism fails to preserve $n$-weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.
Let $(G,d)$ be a metric space with the counting measure $μ$ satisfying some growth conditions. Let $ω(x,y)=(1+d(x,y))^δ$ for some $0<δ\leq1$. Let $0<p\leq1$. Let $\mathcal A_{pω}$ be the collection of kernels $K$ on $G\times G$ satisfying $\max\{\sup_x\sum_y |K(x,y)|^pω(x,y)^p, \sup_y\sum_x |K(x,y)|^pω(x,y)^p\}<\infty$. Each $K \in \mathcal A_{pω}$ defines a bounded linear operator on $\ell^2(G)$. If in addition, $ω$ satisfies the weak growth condition, then we show that $\mathcal A_{pω}$ is inverse closed in $B(\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of inf…
A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints
Purdue University（普渡大学）；University of Vienna（维也纳大学）
We prove that an arbitrary Banach couple is uniquely determined by the collection of intermediate spaces that are interpolation spaces for the operators of rank one. From this we deduce a confirmation to the conjecture by Yu. A. Brudnyi and N.Ya. Kruglyak that a Banach couple is uniquely determined by the collection of all its interpolation spaces. Some relevant problems are also analized.
The notions of module pseudo-amenable and module pseudo-contractible Banach algebras are introduced. For a Banach algebra with bounded approximate identity, module pseudo-amenability and module approximate amenability are the same properties. It is given a complete characterization of module pseudo-amenability for a Banach algebra. For every inverse semigroup $S$ with subsemigroup $E$ of idempotents, necessary and sufficient conditions are obtained for the $\ell^1(S)$ and its second dual to be $\ell^1(E)$-module pseudo-amenable.
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.
