Gruss inequality for some types of positive linear maps

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 matrix algebras by using some matrix tricks. We also give a Gr\"uss operator inequality in the setting of $C^*$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.Comment: 17 pages, to appear in J. Operator Theory (JOT

Eigenvalue extensions of Bohr's inequality

We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr's inequality due to Vasi\'c and Ke\v{c}ki\'c.Comment: 8 pages, to appear in Linear Algebra App

Non-commutative Callebaut inequality

We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices.Comment: 10 pages; to appear in Linear Algebra Appl. (LAA

Some inequalities for unitarily invariant norm

AbstractWe shall prove the inequalities|||(A+B)(A+B)∗|||⩽|||AA∗+BB∗+2AB∗|||⩽|||(A-B)(A-B)∗+4AB∗|||for all n×n complex matrices A,B and all unitarily invariant norms ∣∣∣·∣∣∣. If further A,B are positive definite it is proved that∏j=1kλj(A♯αB)⩽∏j=1kλj(A1-αBα),1⩽k⩽n,0⩽α⩽1,where ♯α denotes the operator means considered by Kubo and Ando and λj(X), 1⩽j⩽n, denote the eigenvalues of X arranged in the decreasing order whenever these all are real. A number of inequalities are obtained as applications