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We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality. An eigenvalue extension of Bohrs inequality is discussed as well.
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We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohrs inequality due to Vasic and Kev{c}kic.
In the spirit of Grothendiecks famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Frechet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optim
Yuan and Leng (2007) gave a generalization of Ky Fans determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(det (A+B))^{1/n} ge (det A)^{1/n} +(det B)^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Lengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu [Linear Algebra Appl. 508 (2016) 206--213].
The arithmetic complexity $c(mathscr{A}_{theta})$ of a noncommutative torus $mathscr{A}_{theta}$ measures the rank $r$ of a rational elliptic curve $mathscr{E}(K)cong mathbf{Z}^r oplus mathscr{E}_{tors}$ via the formula $r= c(mathscr{A}_{theta})-1$. The number $c(mathscr{A}_{theta})$ is equal to the dimension of a connected component $V_{N,k}^0$ of the Brock-Elkies-Jordan variety associated to a periodic continued fraction $theta=[b_1,dots, b_N, overline{a_1,dots,a_k}]$ of the period $(a_1,dots, a_k)$. We prove that the component $V_{N,k}^0$ is a fiber bundle over the Fermat-Pell conic $mathscr{Q}$ with the structure group $mathscr{E}_{tors}$ and the fiber an $r$-dimensional affine space. As an application, we evaluate the Tate-Shafarevich group $W (mathscr{E}(K))$ of elliptic curve $mathscr{E}(K)$ in terms of the group $W (mathscr{Q})$ calculated by Lemmermeyer.
In this article, we explore the celebrated Gr{u}ss inequality, where we present a new approach using the Gr{u}ss inequality to obtain new refinements of operator means inequalities. We also present several operator Gr{u}ss-type inequalities with applications to the numerical radius and entropies.
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