On the norm of polynomials of two adjoint projections (Q1175047)

Let \(f(X,Y)\) denote a polynomial of two non-commuting variables, and \(P\) a nontrivial projection (i.e. \(P=P^ 2\neq 0,1\)) in a Hilbert space \(Z\). The authors study the polynomials \(f(X,Y)\) such that \(\| f(P,P^*)\|\) depend only on \(\| P\|\), and get an explicit function \(\varphi(z)\) with \(\| f(P,P^*)\|=\varphi(\| P\|)\) for such \(f(X,Y)\). As an application, the norms of some singular integral operators are determined. Furthermore, the following facts are also pointed out: 1. In case of Banach space \(Z\), unless \(Z\) is Hilbert space, \(\| 1- P\|\) is never known only by \(\| P\|\), 2. For a polynomial of three non-commuting variables, a simple example shows that the situation is hopeless, 3. \(s_ n(f(P,P^*))=s_ n(f(P^*,P))\) holds for every projection \(P\), where \(s_ n\) means the \(n\)-th \(s\)-number.