of $\\mathscr{A}$‎. ‎In this talk‎, ‎we present several versions of the Haagerup--Pisier--Ringrose inequality involving unitarily invariant norms and unital completely positive maps‎. ‎Among other things‎, ‎we show that if $\\mathcal{J}$ is the ideal...
|\\cdot...
|$‎, ‎$\\Phi‎: ‎\\mathscr{A} \\to \\mathscr{B}$ is a bounded linear map between $C^*$-algebras‎, ‎$A_1‎, ‎\\cdots‎, ‎A_n\\in \\mathscr{A}$ are positive such that $\\Phi(A_j)$'s commute for $1\\leq j\\leq n$ and $X_1‎, ‎\\cdots‎, ‎X_n \\in \\mathcal{J}$‎. ‎Then‎

‎\\begin{align*}‎

‎\\left|\\left|\\left| \\sum_{j=1}^n\\left\\{ \\Phi(A_...
|X_1 \\oplus \\cdots\\oplus X_n...
|\\‎, ‎\\|\\Phi\\|^2\\left\\|\\sum_{j=1}^n A_j^2\\right\\|‎.

‎\\end{align*}‎...

The celebrated Heinz inequality asserts that $

2

Assuming a unitarily invariant norm $...
|\\cdot...
|$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $...
|\\cdot...
|$ on matrix algebras $\\mathcal{M}_n$ for all finite values of $n$ via $...
|A...
|=...
|A\\oplus 0...
|$. We show that if $\\mathscr{A}$ is a $C^*$-algebra of finite dimension

$k$ and $\\Phi: \\mathscr{A} \\to \\mathcal{M}_n$ is a unital completely

positive map, then

\\begin{equation*}

...
|\\Phi(AB)-\\Phi(A)\\Phi(B)...
| \\leq \\frac{1}{4}

...
|I_{n}...
|\\,...
|I_{kn}...
| d_A d_B

\\end{equation*}

for any $A,B \\in

\\mathscr{A}$, where $d_X$ denotes the diameter of the unitary orbit

$\\{UXU^*: U \\mbox{ is unitary}\\}$ of $X$ and $I_{m}$ stands for the

identity of $\\mathcal{M}_{m}$. Further we get an analogous

inequality for certain $n$-positive maps in the setting of full...

We utilize some $2 \\times 2$ matrix tricks to obtain several unitarily invariant norm inequalities corresponding to the L\\"{o}wner--Heinz inequality, the arithmetic--geometric mean inequality and the Corach--Porta--Recht inequality...

We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the ...

In this paper, motivated by perturbation theory of operators, we present some upper bounds for $...
|f(A)Xg(B)+ X...
|$ in terms of $...
|\\,|AXB|+|X|\\,...
|$ and $...
|f(A)Xg(B)- X...
|$ in terms of $...
|\\,|AX|+|XB|\\,...
|$, where $A, B$ are $G_{1}$ operators, $...
|\\cdot...
|$ is a unitarily invariant norm and $f, g$ are certain analytic functions. Further, we find some new upper bounds for the the Schatten $2$-norm of $f(A)X\\pm Xg(B)$. Several special cases are discussed as well....

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\\in {\\mathfrak B...

In this article, we present several inequalities treating operator means and the Cauchy-Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of ...

problem to find the best possible refinement‎. ‎As applications‎, ‎we present matrix versions including unitarily invariant norms‎, ‎trace and determinant versions‎....

present a refinement of the H-P-R inequality and give an example to support it‎.
‎Moreover‎, ‎we establish several versions of the H-P-R inequality involving unitarily invariant norms‎. ‎Further‎, ‎we present some useful inequalities in the setting...