Constrained von Neumann inequalities (Q1599838)

Let \(S^*\) be the backward shift operator on \(\ell^2\), that is, \(S^*(x_0,x_1)=(x_1,x_2,\ldots)\). An equivalent version of the classical von Neumann inequality says that for every Hilbert space contraction \(T\) and every polynomial \(p\) we have \(\| p(T)\| _{{\mathcal B}(H)}\leq \| p(S^*)\| _{{\mathcal B}(\ell^2)}\); that is, \(S^*\) is extremal. The paper under review generalizes these type of inequalities. To this end, the notion of admissible operator radius is introduced; this is a map \(w:{\mathcal B}(H)\to [0,\infty]\) such that (i) \(w(U^*TU)=w(T)\) for each unitary \(U\) and \(T\in {\mathcal B}(H)\); (ii) if \(E\) is an invariant subspace for \(T\in{\mathcal B}(H)\), then \(w(T| _E)\leq w(T);\) and (iii) if \(T^{(\infty)}\) denotes the countable orthogonal sum \(T\oplus T\oplus\dots\), then \(w(T^{(\infty)})=w(T)\). The results obtained then roughly read as follows: if \(T\) is a Hilbert space contraction satisfying some constraints and \({\omega}\) is an admissible operator radius, then there exists an invariant subspace \(E\) of \(S^*\) such that for every polynomial \(p\) one has \(\omega(p(T))\leq \omega(p(S^*| _E))\). As one of many interesting applications of their results, the following estimates for the numerical radius \(\omega_2(T)=\sup\{| \langle Tx,x\rangle| : x\in H\), \(\| x\| =1\}\) of \(T\) are given: If \(T\in {\mathcal B}(H)\) satisfies \(\| T\| \leq 1\), \(T^3\geq 0\) and \(I-2T^*T+T^{*2}T^2=0\), then \(\omega_2(T)\leq \sqrt{7/24}\) and \(\omega_2(T^2)\leq \sqrt{1/12}\), where the constants are the best possible. Further applications to harmonic analysis involve generalizations of coefficient inequalities for positive trigonometric polynomials due to Fejer, Egerváry and Százs.