Square functions and \(H^\infty\) calculus on subspaces of \(L^p\) and on Hardy spaces (Q2574931)

Let \(X\) be a closed subspace of \(L^p\) with \(1\leq p<\infty\). Let \(A\) be any sectorial operator on \(X\). For suitable bounded holomorphic functions \(F\) which decay at 0 and \(\infty\), the associated square function on \(X\) is defined by \[ \|x\|_F=\left\|\left(\int^\infty_0 \bigl| F(tA)x\bigr|^2\frac {dt}{t} \right)^{1/2}\right \|_{L^p}. \] The authors show that if \(A\) has a bounded \(H^\infty\) functional calculus on \(X\), then the square function norms are equivalent to the original norm of \(X\) and obtain a similar result when \(X\) is the usual Hardy space on an Euclidean space. This article extends previously known results when \(X\) is an \(L^p\) space.