A lifting theorem and analytic operator algebras

Abstract:

Let $K$ be a complex Hilbert space and $H$ a closed subspace. It is shown that if a $2 \times 2$ selfadjoint operator matrix $T$ with positive diagonals on $K \oplus K$ is positive on $H \oplus {H^ \bot }$, then there exists a $2 \times 2$ operator matrix $\tilde T$ with the same diagonals such that $\tilde T$ is positive on $K \oplus K$ and $T$ is the restriction of $\tilde T$ to $H \oplus {H^ \bot }$. When $T$ is in a von Neumann algebra, we consider the problems of finding $T$ in the same algebra. This lifting theorem has applications to weighted norm inequalities for conjugation operators on analytic operator algebras.