BKW-operators for Chebyshev systems (Q1970881)

Let \(I=[0,1]\subset\mathbb{R}\) be the closed unit interval and let \(C(I,\mathbb{R})\) be the Banach space of all real valued continuous functions on \(I\). For \(f\in C(I,\mathbb{R})\) let \(\|f\|_0= \sup\{|f(t) |:t\in I\}\). A bounded linear operator \(T\) on \(C(I,\mathbb{R})\) is called a BKW-operator for the Chebyshev system \(S_k=\{1,t,\dots,t^k\}\) if \((T_n)\) is a sequence of bounded linear operators on \(C(I,\mathbb{R})\) such that \(\|T_n\|\to\|T\|\) and \(\|T_nh-Th \|_0\to 0\) for each \(h\in S_k\), then \(\|T_ng-Tg\|_0\to 0\) for every \(g\in C(I,\mathbb{R})\), that is \((I_n)\) converges strongly to \(T\). The authors characterize BKW-operators on \(C(I,\mathbb{R})\) for the Chebyshev system \(S_3\). It is also investigated when subtraction of composition operators are BKW-operators for the Chebyshev system \(S_4\).