The Distinguished Boundary of the Unit Operator Ball

Proof. Consider operators of the form A ©1#jl where A7 is a finite dimensional subspace of H, A is an operator on N with ||.4]| Sil, and lif± is the identity on NL. Such operators are clearly dense in (B in the strong operator topology, and Z7©ljvx is unitary if U is unitary on N. Therefore it is enough to prove (1) for finite dimensional spaces. In that case (1) follows from work of Bochner [l], or alternatively it may be proved as follows. If A is an «X« matrix with ¡mi â 1 we may write A in polar form as A = VP where V is unitary and P is diagonal with entries Xi( 0^X¿^1. Consider the set of all matrices VQ where Q is diagonal with entries f,-, | fi| á¡l. For fixed i, and for fixed values of Xi (j^i), f(VQ) is a holomorphic function of £",-. Apply the maximum modulus theorem n times to get |/C4)| = |/( VQ) |, where Q is diagonal with entries f,on the unit circle, and thus conclude that U= VQ is unitary. As a particular case, let p be a polynomial in one variable with complex coefficients, p(z)=cQ+Ciz+ ■ ■ • +ckzk, such that