Mohammad R. Farmani
In this paper, we show that the essentiality of the scole of an ideal B i a semi-simple Banach algebra A implies that any invertibility preserving isomorphism on A is a Jordan homomorphism. Specially ...
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We prove a normalized version of the restricted invertibility principle obtained by Spielman-Srivastava. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate. As a consequence, we also recover the best known estimate for the Banach-Mazur distance to the cube: the distance of every n-dimensional normed space from \ell_{\infty}^n is at most (2n)^(5/6). Finally, using tools from the work of Batson-Spielman-Srivastava, we give a new proof for a theorem of Kashin-Tzafriri on the norm of restricted matrices.
On generalized core-EP invertibility in a Banach algebra
本文在Banach *-代数中研究广义core-EP逆，利用零化子刻画其性质，得到乘积的广义core-EP逆，并研究core-EP序及其在分块复矩阵中的应用。
We investigate the Hirano invertibility of block-operator matrices in Banach algebras, and obtain the Hirano inverse of matrix $\begin{bmatrix} A&B\\ C&D \end{bmatrix}$ under two types of new perturbation conditions. Furthermore, we provide a new operator matrix decomposed into the sum of tripotent and nilpotent elements on Banach spaces.
Banach algebra mappings preserving the invertibility of linear pencils
本文证明了在复单位Banach代数中，若半单代数具有本质socle，且两个满射映射双向保持线性束的可逆性，则它们可表示为同一Jordan同构与一个可逆元素的乘积。
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…
We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.
The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras
本文研究Banach代数中元素之和的广义与伪n-强Drazin逆，建立可逆性充要条件并证明加法性质，推广已有结果，同时定义并刻画加权版本。
We provide sufficient conditions for a mapping between two Banach spaces to be a diffeomorphism using the approach of an auxiliary functional and also by the aid of a duality mapping corresponding to a normalization function. We simplify and generalize our previous results. Applications to algebraic equations and to integro-differential systems are also given.
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 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.
On the spectral identities and fundamental properties of one-sided Drazin inverses in Banach algebras
本文研究了Banach代数中单边（广义）Drazin逆的基本性质，引入单边强pi-正则性并证明其等价于单边Drazin可逆性，进而利用Jacobson引理推导出单边Drazin可逆谱及Fredholm型算子的谱恒等式。
We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, …
Drazin and g-Drazin invertibility of combinations of three Banach algebra elements
本文研究复含单位元Banach代数中元素组合的Drazin与g-Drazin可逆性，证明了三元素及两元素满足特定条件时的可逆性传递关系，给出幂等元相关组合的可逆性等价结论及逆的表示公式。
We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.
We introduce a concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for t…
On Classes of Fredholm Type Operators
本文在 Banach 空间的 Calkin 代数中研究了 p-可逆性及广义 Drazin 可逆性，进而定义并刻画了 P-Fredholm、伪 B-Fredholm 和弱 B-Fredholm 等算子类。
We show a Kalton-Weis type theorem for the general case of non-commuting operators. More precisely, we consider sums of two possibly non-commuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible, and moreover sectorial. As a…
Additive results of group inverses in Banach algebras
本文研究Banach代数中两个群可逆元素之和的群逆表示，并将所得结果应用于分块复矩阵，从而得到某些分块复矩阵的群可逆性。
Let $\mathscr{X}$ be a complex Banach space and $\mathcal{L}(\mathscr{X})$ be the algebra of all bounded linear operators on $\mathscr{X}$. For a given elementary operator $Φ$ of length $2$ on $\mathcal{L}(\mathscr{X})$, we determine necessary and sufficient conditions for the existence of a solution of the equation ${\rm X} Φ=0$ in the algebra of all elementary operators on $\mathcal{L}(\mathscr{X})$. Our approach allows us to characterize some invertible elementary operators of length $2$ whose inverses are elementary operators.
We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall.
Compact disjointness preserving operators on Banach $C(K)$-modules
将Banach格上紧保不交算子的经典结果推广到有限生成Banach $C(K)$-模，通过模结构刻画算子紧性与保不交性的关系。
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.
From geometry to invertibility preservers
该研究刻画了矩阵空间与算子代数上，双向保持矩阵或算子对之差的满秩及可逆性的双射，属于代数与泛函分析交叉领域的可逆性保持问题研究。
We give an example of a unital commutative complex Banach algebra having a normalized state which is not a spectral state and admitting an extreme normalized state which is not multiplicative. This disproves two results by Golfarshchi and Khalilzadeh.
Let $A$ and $B$ be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ σ(p(Tf,Tg))=σ(p(f,g)), \quad f,g \in A \] holds is an algebra isomorphism.
Interpolation between domains of powers of operators in quaternionic Banach spaces
本文在四元数Banach空间中，利用K-插值方法证明了算子幂的定义域之间的插值关系，并通过伪S-预解算子的可积条件刻画了插值空间。
We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extens…
Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, …
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any…
Let $A$ and $B$ be unital semisimple commutative Banach algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $σ(TfTg)=σ(fg)$; (2) $σ_π(TfTg-α)\cap σ_π(fg-α)\ne \emptyset$; (3) $\mathrm{r} (TfTg-α)=\mathrm{r}(fg-α)$ hold for some non-zero complex number $α$ and for every $f, g\in A^{-1}$, where $σ(\cdot)$ (resp. $σ_π(\cdot)$) denotes the (resp. peripheral) spectrum and $\rr(\cdot)$ denotes the spectral radius. Under each of the hypotheses we show representations for $T$ and unde…
For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $…
Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $Φ:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $Φ(a)Φ(b)=Φ(b)Φ(a)=0$ whenever $ab=ba=0$). We show that $Φ$ is a weighted Jordan homomorphism provided that $A$ is zero product determined and weakly amenable. These conditions are in particular fulfilled when $A$ is the group algebra $L^1(G)$ with $G$ any locally compact group. We also study a more general type of continuous linear maps $Φ:A\to B$ that satisfy $Φ(a)Φ(b)+Φ(b)Φ(a)=0$ whenever $ab=ba=0$. We show in particul…
We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form $T=wU$, where $w$ is a central operator and $U$ is a disjointness preserving operator such that its spectrum $σ(U)$ is a subset of the unit circle.
Order-preserving unique Hahn-Banach extensions
本文研究带单位元的实Banach格中子空间的序与等距结构，通过紧凸集及Choquet单纯形的相关条件，得出正泛函延拓唯一性对应的集合特性结论。
We study the invertibility of Banach algebras elements in their extensions, and invertible extensions of Banach and Hilbert space operators with prescribed growth conditions for the norm of inverses. As applications, the solutions of two open problems are obtained. In the first one we give a characterization of E(T)-subscalar operators in terms of growth conditions. In the second one we show that operators satisfying a Beurling-type growth condition possess Bishop's property beta. Other applications are also given.
Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E \to F$, $y \in Y$, and a function $h: Y \to X$. In this paper, we consider maps having the property: \cap^{k}_{i=1}Z(f_{i}) \neq\emptyset\iff\cap^{k}_{i=1}Z(Tf_{i}) \neq \emptyset, where $Z(f) = \{f = 0\}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces…
We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space $\ell_p$, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach sp…
Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\inω}$ in $A$ and unconditionally convergent series $\sum_{n\inω}x_n$ in $X$ the series $\sum_{n\inω}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining …
The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces
研究闭算子T在广义逆T^+存在时，线性扰动δT使得T+δT保持广义逆稳定的条件，推广了Huang在Linear Algebra and its Applications中的主要结果。
We prove the Cauchy-Rassias stability of linear n-inner product preserving mappings in $n$-inner product Banach spaces. We apply the Cauchy-Rassias inequality that plays an influencial role in the subject of functional equations. The inequality was introduced for the first time by Th.M.Rassias in his paper entitled: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.Soc. 72(1978), 297-300.
Let B(X) be the algebra of all bounded linear operators on a complex Banach space X with dim X greater than 3. In this paper, we characterize the forms of surjective linear maps on B(X) which preserve the dimension of the vector space containing of all fixed points of operators, whenever X is a fifinite dimensional Banach space. Moreover, we characterize the forms of linear maps on B(X) which preserve the vector space containing of all fixed points of operators.
In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship betw…
Polyhedral norms and smooth Hahn-Banach extension
本文在多面体范数下，给出了子空间上的光滑泛函存在保范光滑延拓到全空间的充要条件，即该光滑泛函的唯一绝对范数达到点是子空间单位球和全空间单位球的极点，并证明了该结果在非多面体范数下不成立。
We give a characterization of conditional expectation operators through a disjointness type property similar to band preserving operators. We say that the operator $T:X\to X$ on a Banach lattice $X$ is semi band preserving if and only if for all $f, g \in X$, $f \perp Tg$ implies that $Tf \perp Tg$. We prove that when $X$ is a purely atomic Banach lattice, then an operator $T$ on $X$ is a weighted conditional expectation operator if and only if $T$ is semi band preserving.
On symmetric and approximately symmetric operators
本文引入局部保正交算子研究右对称性，证明光滑自反Banach空间上的光滑紧算子若非秩一则非右对称，并刻画有限维空间上近似保正交算子的单射性。
Commutativity preserving mappings in Banach algebras
研究Banach代数中保持交换性的映射，证明在特定条件下，此类映射可分解为同态与反同态的和，结合中心元和加法映射。
Let $E$ be a real Banach space. For $x,y \in E,$ we follow R.James in saying that $x$ is orthogonal to $y$ if $\|x+αy\|\geq \|x\|$ for every $α\in R$. We prove that every operator from $E$ into itself preserving orthogonality is an isometry multiplied by a constant.
Let $E$ and $F$ be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice $E$, which shows that in this case the unbounded disjointness operators from $E\to F$ separate the points of $E$. Then we show that every disjointness preserving operator $T:E\to F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T:E\to F$ has a unique decomposition $T=R+S$, where $R$ is a bounded disjointness preserving operator and…
A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on $\mathbb{R}^n$ is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and…
Resolving the generalized hyperbolicity conjecture for shadowing
该研究通过构造反例证明，关于跟踪性的广义双曲性猜想在一般巴拿赫空间上不成立，但在可分希尔伯特空间上成立，明确了两类空间的差异源于满射与右可逆的间隙。
Let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $ϕ$ on ${\mathcal B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\to X$ and $V:X\to X$ both linear or both conjugate linear such that $ϕ(T)=UTV$ for all $T\in{\mathcal B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\to X$ and $V:X\to X^*$ both linear or both conjugate linear such that $ϕ(T)=UT^*V$ for all $T\in{\mathcal B}(X)$. As immediate consequences of t…
We study the stability of band preserving operators on Banach lattices. To this end the notion of $\varepsilon$-band preserving mapping is introduced. It is shown that, under quite general assumptions, a $\varepsilon$-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of $\varepsilon$-band preserving maps are also obtained.
We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$.
Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.
Let $\cal A$ and $\cal B$ be Banach algebras. A linear map $T:{\cal A} \rightarrow {\cal B}$ is called separating or disjointness preserving if $ab=0$ implies $Ta\;Tb = 0$ for all $a,b\in {\cal A}$. In this paper, we study a new class of regular Tauberian algebras and prove that some well-known Banach algebras in harmonic analysis belong to this class. We show that a bijective separating map between these algebras turns out to be continuous and the maximal ideal spaces of underlying algebras are homeomorphic. By imposing extra conditions on these algebras, we find a more thorough characterizat…
Let $X$ be a complex Banach space with $\dim X\geq3$ and $B(X)$ the algebra of all bounded linear operators on $X$. Suppose $ϕ:B(X)\longrightarrow B(X)$ is a surjective map satisfying the following property: $Fix(AB)=Fix(ϕ(A)ϕ(B)), (A, B\in B(X))$. Then the form of $ϕ$ is characterized, where $Fix(T)$ is the set of all fixed points of an operator $T$.
We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show.
In this study, we analyze the various strengthening and weakening of the uniqueness of the Hahn--Banach extension. In addition, we consider the case in which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/ (SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We obtain various new characterizations of these properties. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is observed that a hyperplane in $c_0$ has property-$(HB)$ if and only if it is an $M$-summand. Conside…
On T-orthogonality in Banach spaces
该研究引入Banach空间中基于有界线性算子T的T-正交性概念，探究其与空间几何性质的关联，刻画保T-正交的算子并借此刻画Hilbert空间。
We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space $X$ is $\{\pm ϕ^k: ϕ\in X^*, \| ϕ\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\hat\bigotimes^{k,s}_{ε_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{ε_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $…
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