arXiv subjects

J. Rooin

Publications and source records attributed to J. Rooin.

3 recordsLinked to original sources

arXiv2018

Operator Ky Fan type inequalities

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions. Among other things, we prove that if $f:[0,\infty)\rightarrow[0,\infty)$ is an operator monotone function with $f (1) = 1$, $f'(1)=μ$, and associated mean $σ$, then for all operators $A$ and $B$ on a complex Hilbert space $\mathscr{H}$ such that $0<A,B\leq\frac{1}{2}I$, we have \begin{equation*} A'\nabla_μB'-A'σB'\leq A\nabla_μB-AσB, \end{equation*} where $I$ is the identity operator on $\mathscr{H}$, $A':=I-A$, $B':=I-B$, and $\nabla_μ$ is the $μ$-weighted arithmetic mean.

math.FA↗

arXiv2018

Sharp inequalities for the numerical radius of block operator matrices

In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of Kittaneh et al., it is shown that if $T=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}$, and $f$ and $g$ are non-negative continuous functions on $[0,\infty)$ such that $f(t)g(t)=t\,\,(t\geq 0)$, then for all nonnegative nondecreasing convex functions $h$ on $[0,\infty)$ , we obtain that \begin{align*}h\left(w^r(T)\right)\leq \max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|A\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|A^*\right|)\right)\right\|, \left\|\frac{1}{p}h\left(f^{pr}(\left|D\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|D^*\right|)\right)\right\|\right), \end{align*} where $p, q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $r\min(p,q)\geq 2$.

math.FA↗

arXiv2017

Geometric aspects of $p$-angular and skew $p$-angular distances

Corresponding to the concept of $p$-angular distance $α_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-angular distance $β_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x, y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances. We then give some results comparing two different $p$-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $α_p[x,y]\geqβ_p[x,y]$.

math.FA↗