For positive semidefinite matrices A and B, Ando and Zhan proved the inequalities |||f(A) + f(B)||| ≥ |||f(A + B)||| and |||g(A) + g(B)||| ≤ |||g(A + B)|||, for any unitarily invariant norm, and for any non-negative operator monotone f on [0,∞) with inverse function g. These inequalities have very recently been generalised to non-negative concave functions f and non-negative convex functions g,...

Let Ai and Bi be positive definite matrices for all i=1,…,m. It is shown that|||∑i=1m(Ai2♯Bi2)r|||≤|||((∑i=1mAi)rp2(∑i=1mBi)rp(∑i=1mAi)rp2)1p|||, unitarily invariant norms, where p>0 r≥1 such that rp≥1. This gives an affirmative answer to a conjecture posed by Dinh, Ahsani Tam. The preceding inequality directly leads recent result of Audenaert in 2015.

For a large class of unitarily invariant reproducing kernel functions K on the unit ball \(\mathbb B_d\) in C^d\), we characterize K-inner as admitting suitable transfer function realization. We associate with each K-contraction T ∈ L(H)d canonical operator-valued and extend uniqueness theorem Arveson for minimal K-dilations to our setting. thus generalize results Olofsson m-hypercontractions d...

We point out a simple criterion for convergence of polynomials to concrete entire function in the Laguerre-Pólya ( LP ) class (of all functions arising as uniform limits with only real roots). then use this show that any random can be obtained limit rescaled characteristic principal submatrices an infinite unitarily invariant Hermitian matrix. Conversely, matrix converge uniformly function. Thi...

Let ( X , p j stretchy="fa...

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments. The classical Franz-Reidemeister torsion τFR is an invariant of manifolds with acyclic unitarily flat vector bundles [62], [33]. In contrast to most other algebraic-topological invariants known at that time, it is invariant under homeomorp...

we introduce a homological invariant of a manifold known as Gromov’s norm. Gromov’s norm of hyperbolic manifolds will be seen to be proportional to the volume of the manifold. The first striking consequence of this result is that the volume of a hyperbolic manifold is a topological invariant. Intuitively, Gromov’s norm measures the efficiency with which multiples of a homology class can be repr...

For a 3-manifold M , McMullen derived from the Alexander polynomial of M a norm on H(M,R) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of th...