An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies $\Vert u\Vert =\Vert u^{-1}\Vert =1$. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra $\BL(X)$ of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) $\BL(X)$ is unitary and, for Y equal to $X,$ $X^*$ or $X^{**},$ there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y.