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Maryam Amyari

Publications and source records attributed to Maryam Amyari.

5 recordsLinked to original sources
arXiv2020

Some refinements of numerical radius inequalities

In this paper, we give some refinements for the second inequality in $\frac{1}{2}\|A\| \leq w(A) \leq \|A\|$, where $A\in B(H)$. In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot)$, we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}ζ(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|$, where $ζ(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{λ,1-λ\}$ and $0\leq λ\leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any operator $A \in B(H)$ in the case when $n=2$.

math.FA↗
arXiv2020

Approximate numerical radius orthogonality

We introduce the notion of approximate numerical radius (Birkhoff) orthogonality and investigate its significant properties. Let $T, S\in \mathbb{B}(\mathscr{H})$ and $\varepsilon \in [0, 1)$. We say that $T$ is approximate numerical radius orthogonal to $S$ and we write $T\perp^{\varepsilon}_ω S$ if $$ω^2(T+λS)\geq ω^2(T)-2\varepsilon ω(T) ω(λS)\,\,\, \text{for all }λ\in\mathbb{C}.$$ We show that $T\perp^{\varepsilon}_ω S$ if and only if $\displaystyle\inf_{θ\in [0, 2π)} D^θ_ω(T, S) \geq -\varepsilon ω(T) ω(S)$ in which $D^θ_ω(T, S)=\displaystyle\lim_{r\to 0^+} \frac{ω^2(T+re^{iθ} S)-ω^2(T)}{2r}$; and this occurs if and only if for every $θ\in[0,2π)$, there exists a sequence $\{x_n^θ\}$ of unit vectors in $\mathscr{H}$ such that $$\displaystyle\lim_{n\to \infty} |\langle Tx^θ_n, x^θ_n\rangle|=ω(T),\,\, \text{and}\,\, \displaystyle\lim_{n\to \infty} {\rm Re}\{e^{-iθ} \langle Tx^θ_n, x^θ_n\rangle\bar{\langle Sx^θ_n, x^θ_n\rangle}\}\geq -\varepsilon ω(T) ω(S),$$ where $ω(T)$ is the numerical radius of $T$.

math.FA↗
arXiv2020

More on $ω$-orthogonality and $ω$-parallelism

We investigate some aspects of various numerical radius orthogonalities and numerical radius parallelism for bounded linear operators on a Hilbert space $\mathscr{H}$. Among several results, we show that if $T,S\in \mathbb{B}(\mathscr{H})$ and $M^*_{ω(T)}=M^*_{ω(S)}$, then $T\perp_{ωB} S$ if and only if $S\perp_{ωB} T$, where $M^*_{ω(T)}=\{\{x_n\}:\,\,\,\|x_n\|=1, \lim_n|\langle Tx_n, x_n\rangle|=ω(T)\}$, and $ω(T)$ is the numerical radius of $T$ and $\perp_{ωB}$ is the numerical radius Birkhoff orthogonality.

math.FA↗
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.

math.FA↗