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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then ${\lambda }_k\left(\right|ab*\left|\right)\le {\lambda }_k{(1/p|a|}^p+1/{q\left|b\right|}^q)$ for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if ${\left|a\right|}^p={\left|b\right|}^q$.

Let $U_c\left(\right)$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{\infty }$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c\left(\right)$. If $u₀,u₁,u\in U_c\left(\right)$ and the geodesic β joining u₀ and u₁ in $U_c\left(\right)$ satisfy $d_{\infty }(u,\beta )<\pi /2$, then the map $f\left(s\right)=d_{\infty }(u,\beta \left(s\right))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_c\left(\right)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_p\left(\right)$ = u: u unitary and u-1 in the p-Schatten class...

Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, ${u|}_{\partial \Omega }=0$. We show that is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators....