On the Fourier transform for operators on homogeneous Banach spaces (Q1344200)

Let \(B\) be a homogeneous Banach space on the circle group \(\mathbb{T}\), and \(L(B)\) denote the Banach algebra of bounded linear operators on \(B\) with the usual operator norm. For \(T\in L(B)\), let \(\widehat T\) denote the Fourier transform of \(T\) (\(\widehat T\) is an \(L(B)\)-valued function on \(\mathbb{Z}\)). We denote by \(\sigma_n(T)\) and \(S_n(T)\) the \(n\)th \(C\)-\(1\) sum and the \(n\)th partial sum of the Fourier series of \(T\), respectively. For \(0< \alpha\leq 1\), the authors introduced the Lipschitz class \(\text{Lip}_\alpha(B)\) in \(L(B)\) and obtained the following results. Theorem 1. If \(T\in \text{Lip}_\alpha(B)\), then \[ |\sigma_n(T)- T|= \begin{cases} O(n^{- \alpha})\quad & (0< \alpha< 1),\\ O(n^{- 1}\log n)\quad & (\alpha= 1).\end{cases} \] Theorem 2. If \(T\in \text{Lip}_\alpha(B)\), then \[ |S_n(T)- T|= O(n^{- \alpha} \log n). \] In particular, the Fourier series of \(T\) converges to \(T\) in the operator norm.