Invertibility of the commutator of an element in a C*-algebra and its Moore-Penrose inverse

Julio Benítez; Vladimir Rakočević

Displaying similar documents to “Invertibility of the commutator of an element in a C*-algebra and its Moore-Penrose inverse”

On the classification of inverse limits of tent maps

Louis Block, Slagjana Jakimovik, Lois Kailhofer, James Keesling (2005)

Fundamenta Mathematicae

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Let $f_{s}$ and $f_{t}$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_{s}$ and $f_{t}$ are periodic and the inverse limit spaces $(I, f_{s})$ and $(I, f_{t})$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

Weighted $w$ -core inverses in rings

Liyun Wu, Huihui Zhu (2023)

Czechoslovak Mathematical Journal

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Let $R$ be a unital $*$ -ring. For any $a, s, t, v, w \in R$ we define the weighted $w$ -core inverse and the weighted dual $s$ -core inverse, extending the $w$ -core inverse and the dual $s$ -core inverse, respectively. An element $a \in R$ has a weighted $w$ -core inverse with the weight $v$ if there exists some $x \in R$ such that $a w x v x = x$ , $x v a w a = a$ and ${(a w x)}^{*} = a w x$ . Dually, an element $a \in R$ has a weighted dual $s$ -core inverse with the weight $t$ if there exists some $y \in R$ such that $y t y s a = y$ , $a s a t y = a$ and ${(y s a)}^{*} = y s a$ . Several characterizations of weighted $w$ -core invertible and weighted dual $s$ -core...

The inverse problem in the calculus of variations: new developments

Thoan Do, Geoff Prince (2021)

Communications in Mathematics

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We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for $n = 2$ . We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.

Linear maps preserving $A$ -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let $ℋ$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ (ℋ)$ and $ℬ_{A} (ℋ)$ the sub-algebra of $ℬ (ℋ)$ of all $A$ -self-adjoint operators. Assume $φ : ℬ_{A} (ℋ)$ onto itself is a linear continuous map. This paper shows that if $φ$ preserves $A$ -unitary operators such that $φ (I) = P$ then $ψ$ defined by $ψ (T) = P φ (P T)$ is a homomorphism or an anti-homomorphism and $ψ (T^{♯}) = ψ {(T)}^{♯}$ for all $T \in ℬ_{A} (ℋ)$ , where $P = A^{+} A$ and $A^{+}$ is the Moore-Penrose inverse of $A$ . A similar result is also true if $φ$ preserves $A$ -quasi-unitary operators in both directions such that there...

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group $_{f} \subseteq G * H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_{f} {= p |}_{f} :_{f} \to G$ . A right inverse (section) $G \to_{f}$ of $p_{f}$ is called a coaction on G. In this paper we study $_{f}$ and the sections of $p_{f}$ . We consider the following topics: the structure of $_{f}$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

$(m, r)$ -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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For integers $m > r \geq 0$ , Brietzke (2008) defined the $(m, r)$ -central coefficients of an infinite lower triangular matrix $G = (d, h) = {(d_{n, k})}_{n, k \in ℕ}$ as $d_{m n + r, (m - 1) n + r}$ , with $n = 0, 1, 2, \dots$ , and the $(m, r)$ -central coefficient triangle of $G$ as $G^{(m, r)} = {(d_{m n + r, (m - 1) n + k + r})}_{n, k \in ℕ} .$ It is known that the $(m, r)$ -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G = (d, h)$ with $h (0) = 0$ and $d (0), h^{'} (0) \neq 0$ , we obtain the generating function of its $(m, r)$ -central coefficients and give an explicit representation for the $(m, r)$ -central Riordan array $G^{(m, r)}$ in terms of the Riordan array $G$ ....

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim (2022)

Archivum Mathematicum

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Let ${U_{n}} = {U_{n} (P, Q)}$ and ${V_{n}} = {V_{n} (P, Q)}$ be the Lucas sequences of the first and second kind respectively at the parameters $P \geq 1$ and $Q \in {- 1, 1}$ . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation $x^{2} - 3 x y + y^{2} + x = 0,$ where $(x, y) = (U_{i}, U_{j})$ or $(V_{i}, V_{j})$ with $i$ , $j \geq 1$ . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

Continuous images of Lindelöf $p$ -groups, $σ$ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that there exists a $σ$ -compact topological group which cannot be represented as a continuous image of a Lindelöf $p$ -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$ -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$ -group, then there exists a covering $γ$ of $Y$ by dyadic compacta such that $| γ | \leq 2^{ω}$ . We also show that if a homogeneous compact space $Y$ is...

A new Lindelöf space with points $G_{δ}$

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that $♦^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{ℵ_{1}}$ which has points $G_{δ}$ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.