If H is a Hilbert space, A is a positive bounded linear operator on H and S is a closed subspace of H, the relative position between S and A−1(S⊥) establishes a notion of compatibility. We show that the compatibility of (A,S) is equivalent to the existence of a convenient orthogonal projection in the operator range R(A1/2) with its canonical Hilbertian structure.

We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete Kσ . We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equ...

Recently the author proved that Hummel–Scheinberg–Zalcman conjecture of 1977 on coefficients nonvanishing Hp functions is true for all p = 2m, m ϵ ℕ, i.e., Hilbertian Hardy spaces H2m. As a consequence, this also implies proof Krzyz bounded functions, which originated direction. In present paper, we solve problem with ≥ 2.