The convex properties and norm bounds for operator matrices involving contractions

Abstract

In this note,
the norm bounds and convex properties of special operator matrices $\widetilde{H}_{n}^{(m)}$ and $\widetilde{S}_{n}^{(m)}$ are investigated. When Hilbert space
$\mathcal{K}$ is infinite dimensional, we firstly show that $\widetilde{H}_{n}^{(m)}=\widetilde{H}_{n+1}^{(m)}$ and $\widetilde{S}_{n}^{(m)}=\widetilde{S}_{n+1}^{(m)},$ for $m,n=1,2,\cdots.$ Then we get that
$\widetilde{H}_n^{(m)}$ is a convex and compact set in the $\omega^{*}$ topology.
Moreover, some norm bounds for $\widetilde{H}_n^{(m)}$ and $\widetilde{S}_{n}^{(m)}$ are given.