Lifting properties, Nehari theorem and Paley lacunary inequality (Q1082599)

Let V be a vector space, \(V_ 1,V_ 2\) two subspaces of V, \(\sigma_ 1,\sigma_ 2\) two seminorms in V, L a family of sesquilinear forms \(S: V\times V\to {\mathbb{C}}\) such that each \(S\in L\) satisfies \(| S(a,b)| \leq \sigma_ 1(a)\sigma_ 2(b)\) for all \((a,b)\in V_ 1\times V_ 2\), and let \(\Lambda '=\{S\in \Lambda:| S(a,b)| \leq \sigma_ 1(a)\sigma_ 2(b)\) for all (a,b)\(\in V\times V\}\). \(S\in L\) is said to have a lifting to L' if there exists S'\(\in L'\) with \(S'=S\) on \(V_ 1\times V_ 2\), and L has the lifting property if every \(S\in L\) can be lifted to L'. The main theorem of the paper asserts that L has the lifting property in the case where V is the set of all sequences a:\({\mathbb{Z}}\to {\mathbb{C}}\) of finite support, \(V_ 1=\{a\in V:a(n)=0\) for \(n<0\}\), \(V_ 2=\{a\in V:a(u)=0\) for \(n\geq 0\}\), and where all \(S\in L\) are Toeplitz forms and the \(\sigma_ 1,\sigma_ 2\) satisfy some natural conditions. As applications some refinements and extensions of the Nehari theorem and the Paley lacunary inequality are derived. In forthcoming papers the theorem is extended for the case where \(V_ 1=V_ 2=\{a:\) \({\mathbb{R}}\to {\mathbb{C}}:a(t)=0\) if \(| t| >\rho \}\), and for sesquilinear forms defined in Lax-Phillips evolutions.