arXiv subjects

Ali Zamani

Publications and source records attributed to Ali Zamani.

36 records · Page 2Linked to original sources

arXiv2020

Numerical radius orthogonality in $C^*$-algebras

In this paper we characterize the Birkhoff--James orthogonality with respect to the numerical radius norm $v(\cdot)$ in $C^*$-algebras. More precisely, for two elements $a, b$ in a $C^*$-algebra $\mathfrak{A}$, we show that $a\perp_{B}^{v} b$ if and only if for each $θ\in [0, 2π)$, there exists a state $φ_{_θ}$ on $\mathfrak{A}$ such that $|φ_{_θ}(a)| = v(a)$ and $\mbox{Re}\big(e^{iθ}\overline{φ_{_θ}(a)}φ_{_θ}(b)\big)\geq 0$. Moreover, we compute the numerical radius derivatives in $\mathfrak{A}$. In addition, we characterize when the numerical radius norm of the sum of two (or three) elements in $\mathfrak{A}$ equals the sum of their numerical radius norms.

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arXiv2020

A-numerical radius and product of semi-Hilbertian operators

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arXiv2019

Birkhoff--James orthogonality of operators in semi-Hilbertian spaces and its applications

In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\mathcal{H}$ such that $\displaystyle{\lim_{n\rightarrow +\infty}}{\|Tx_n\|}_A = {\|T\|}_A$ and $\displaystyle{\lim_{n\rightarrow +\infty}}{\langle Tx_n, Sx_n\rangle}_A = 0$. In addition, we give some $A$-distance formulas. Particularly, we prove \begin{align*} \displaystyle{\inf_{γ\in \mathbb{C}}}{\|T + γS\|}_{A} = \sup\Big\{|{\langle Tx, y\rangle}_A|; \, {\|x\|}_{A} = {\|y\|}_{A} = 1, \, {\langle Sx, y\rangle}_A = 0\Big\}. \end{align*} Some other related results are also discussed.

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arXiv2019

$A$-numerical radius inequalities for semi-Hilbertian space operators

Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A$ and $w_A(T)$ denote the $A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In this paper, we prove the following characterization of $w_A(T)$ \begin{align*} w_A(T) = \displaystyle{\sup_{α^2 + β^2 = 1}} {\left\|α\frac{T + T^{\sharp_A}}{2} + β\frac{T - T^{\sharp_A}}{2i}\right\|}_A, \end{align*} where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. We then apply it to find upper and lower bounds for $w_A(T)$. In particular, we show that \begin{align*} \frac{1}{2}{\|T\|}_A \leq \max\Big\{\sqrt{1 - {|\cos|}^2_AT}, \frac{\sqrt{2}}{2}\Big\}w_A(T)\leq w_A(T), \end{align*} where ${|\cos|}_AT$ denotes the $A$-cosine of angle of $T$. Some upper bounds for the $A$-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given.

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arXiv2018

Orthogonally $a$-Jensen mappings on $C^*$-modules

We investigate the representation of the so-called orthogonally $a$-Jensen mappings acting on $C^*$-modules. More precisely, let $\mathfrak{A}$ be a unital $C^*$-algebra with the unit $1$, let $a \in \mathfrak{A}$ be fixed such that $a, 1-a$ are invertible and let $\mathscr{E}, \mathscr{F}, \mathscr{G}$ be inner product $\mathfrak{A}$-modules. We prove that if there exist additive mappings $φ, ψ$ from $\mathscr{F}$ into $\mathscr{E}$ such that $\big\langle φ(y), ψ(z)\big\rangle=0$ and $a \big\langle φ(y), φ(z)\big\rangle a^\ast = (1 - a)\big\langle ψ(y), ψ(z)\big\rangle (1 - a)^\ast$ for all $y, z\in \mathscr{F}$, then a mapping $f: \mathscr{E} \to \mathscr{G}$ is orthogonally $a$-Jensen if and only if it is of the form $f(x) = A(x) + B(x, x) +f(0)$ for $x\in \mathscr{K} := φ(\mathscr{F})+ψ(\mathscr{F})$, where $A: \mathscr{E} \to \mathscr{G}$ is an $a$-additive mapping on $\mathscr{K}$ and $B$ is a symmetric $a$-biadditive orthogonality preserving mapping on $\mathscr{K}\times \mathscr{K}$. Some other related results are also presented.

math.OA↗

arXiv2018

Approximately angle preserving mappings

In this paper, we present some characterizations of linear mappings, which preserve vectors at a specific angle. We introduce the concept of $(\varepsilon, c)$-angle preserving mappings for $|c|<1$ and $0\leq \varepsilon < 1 + |c|$. In addition, we define $\widehat{\varepsilon}\,(T, c)$ as the ``smallest'' number $\varepsilon$ for which $T$ is $(\varepsilon, c)$-angle preserving mapping. We state some properties of the function $\widehat{\varepsilon}\,(., c)$, and then propose an exact formula for $\widehat{\varepsilon}\,(T, c)$ in terms of the norm $\|T\|$ and the minimum modulus $[T]$ of $T$. Finally, we characterize the approximately angle preserving mappings.

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arXiv2018

Numerical radius parallelism of Hilbert space operators

In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\big(\mathscr{H}, \langle \cdot ,\cdot \rangle\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $ω(T + λS) = ω(T)+ω(S)$ for some complex unit $λ$. We show that $T \parallel_ω S$ if and only if there exists a sequence of unit vectors $\{x_n\}$ in $\mathscr{H}$ such that \begin{align*} \lim_{n\rightarrow\infty} \big|\langle Tx_n, x_n\rangle\langle Sx_n, x_n\rangle\big| = ω(T)ω(S). \end{align*} We then apply it to give some applications.

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arXiv2018

Stable Stair-Climbing of a Quadruped Robot

Synthesizing a stable gait that enables a quadruped robot to climb stairs is the focus of this paper. To this end, first a stable transition from initial to desired configuration is made based on the minimum number of steps and maximum use of the leg workspace to prepare the robot for the movement. Next, swing leg and body trajectories are planned for a successful stair- climbing gait. Afterwards, a stable spinning gait is proposed to change the orientation of the body. We simulate our gait planning algorithms on a model of quadruped robot. The results show that the robot is able to climb up stairs, rotate about its yaw axis, and climb down stairs while its stability is guaranteed.

cs.RO↗

arXiv2018

Characterizations of norm--parallelism in spaces of continuous functions

In this paper, we consider the characterization of norm--parallelism problem in some classical Banach spaces. In particular, for two continuous functions $f, g$ on a compact Hausdorff space $K$, we show that $f$ is norm--parallel to $g$ if and only if there exists a probability measure (i.e. positive and of full measure equal to $1$) $μ$ with its support contained in the norm attaining set $\{x\in K: \, |f(x)| = \|f\|\}$ such that $\big|\int_K \overline{f(x)}g(x)dμ(x)\big| = \|f\|\,\|g\|$.

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arXiv2018

Switching between Limit Cycles in a Model of Running Using Exponentially Stabilizing Discrete Control Lyapunov Function

This paper considers the problem of switching between two periodic motions, also known as limit cycles, to create agile running motions. For each limit cycle, we use a control Lyapunov function to estimate the region of attraction at the apex of the flight phase. We switch controllers at the apex, only if the current state of the robot is within the region of attraction of the subsequent limit cycle. If the intersection between two limit cycles is the null set, then we construct additional limit cycles till we are able to achieve sufficient overlap of the region of attraction between sequential limit cycles. Additionally, we impose an exponential convergence condition on the control Lyapunov function that allows us to rapidly transition between limit cycles. Using the approach we demonstrate switching between 5 limit cycles in about 5 steps with the speed changing from 2 m/s to 5 m/s.

cs.RO↗

arXiv2018

Characterization of numerical radius parallelism in $C^*$-algebras

Let $v(x)$ be the numerical radius of an element $x$ in a $C^*$-algebra $\mathfrak{A}$. First, we prove several numerical radius inequalities in $\mathfrak{A}$. Particularly, we present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. In addition, we show that if $x\in\mathfrak{A}$, then $v(x) = \frac{1}{2}\|x\|$ if and only if $\|x\| = \|\mbox{Re}(e^{iθ}x)\| + \|\mbox{Im}(e^{iθ}x)\|$ for all $θ\in \mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-algebras based on numerical radius. More precisely, we consider elements $x$ and $y$ of $\mathfrak{A}$ which satisfy $v(x + λx) = v(x) + v(y)$ for some complex unit $λ$. We show that this relation can be characterized in terms of pure states acting on $\mathfrak{A}$.

math.OA↗

arXiv2017

Characterizations of smooth spaces by $ρ_*$-orthogonality

The aim of this paper is to present some results concerning the $ρ_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, ρ_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\leq 3[T]\|x\|, \qquad (x\in X)$$ where $[T]:=\inf\{\|Tx\|: \,x\in X, \|x\|=1\}$. We also show that the pair $(X,\perp_{ρ_*})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the $ρ_*$-orthogonality.

math.FA↗

arXiv2017

Characterizations of operator Birkhoff-James orthogonality

In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{\mathscr{H}_0}^\perp} < \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $ξ\in {\mathscr{H}}_0$ such that $\|T\|ξ= |T|ξ$ and $S^*Tξ= 0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^*$-modules.

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arXiv2016

Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(δ, \varepsilon)$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(δ, \varepsilon)$-orthogonality preserving. In particular, if $\mathscr{E}$ is a full Hilbert $\mathscr{A}$-module with $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T, S:\mathscr{E}\longrightarrow \mathscr{E}$ are two linear mappings satisfying $\big|\langle Sx, Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for all $x, y\in \mathscr{E}$ and $\|T - S\| \leq θ\|S\|$, then we show that $T$ is a $(δ, \varepsilon)$-orthogonality preserving mapping. We also prove whenever $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T: \mathscr{E} \longrightarrow \mathscr{F}$ is a nonzero $\mathscr{A}$-linear $(δ, \varepsilon)$-orthogonality preserving mapping between $\mathscr{A}$-modules, then $$\big\|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle\big\|\leq \frac{4(\varepsilon - δ)}{(1 - δ)(1 + \varepsilon)} \|Tx\|\,\|Ty\|\qquad (x, y\in \mathscr{E}).$$ As a result, we present some characterizations of the orthogonality preserving mappings.

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arXiv2015

Norm-parallelism in the geometry of Hilbert $C^*$-modules

Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators and elements of an arbitrary $C^*$-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^*$-module.

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arXiv2015

Approximately bisectrix-orthogonality preserving mappings

Regarding the geometry of a real normed space ${\mathcal X}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in {\mathcal X}$ as follows: $${x\np{\varepsilon}}_W y \mbox{if and only if} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study class of linear mappings preserving the approximately bisectrix-orthogonality ${\np{\varepsilon}}_W$. In particular, we show that if $T: {\mathcal X}\to {\mathcal Y}$ is an approximate linear similarity, then $${x\npδ}_W y\Longrightarrow {Tx \npθ}_W Ty \qquad (x, y\in {\mathcal X})$$ for any $δ\in[0, 1)$ and certain $θ\geq 0$.

math.FA↗

arXiv2015

Angle Preserving Mappings

In this paper, we give some characterizations of orthogonality preserving mappings between inner product spaces. Furthermore, we study the linear mappings that preserve some angles. One of our main results states that if $\mathcal{X}, \mathcal{Y}$ are real inner product spaces and $θ\in(0, π)$, then an injective nonzero linear mapping $T:\mathcal{X}\longrightarrow \mathcal{Y}$ is a similarity whenever (i) $x\undersetθ{\angle} y\, \Leftrightarrow \,Tx\undersetθ{\angle} Ty$ for all $x, y\in \mathcal{X}$; (ii) for all $x, y\in \mathcal{X}$, $\|x\|=\|y\|$ and $x\undersetθ{\angle} y$ ensure that $\|Tx\|=\|Ty\|$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{*}$-modules. Another result shows that if $\mathbb{K}(\mathscr{H})\subseteq\mathscr{A}\subseteq\mathbb{B}(\mathscr{H})$ is a $C^{*}$-algebra and $T\,:\mathscr{E}\longrightarrow \mathscr{F}$ is an $\mathscr{A}$-linear mapping between inner product $\mathscr{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\leq|y|\, \Rightarrow \,|Tx|\leq|Ty|$ for all $x, y\in \mathscr{E}$.

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arXiv2014

Exact and approximate operator parallelism

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module.

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