A lifting theorem and analytic operator algebras

Abstract

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