Abstract :
An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued
multipliers S(λ) for the reproducing kernel Hilbert space H(kd ) on the unit ball Bd ⊂ Cd, where kd is the positive kernel
kd (λ, ζ ) = 1/(1 − λ, ζ ) on Bd . The reproducing kernel space H(KS) associated with the positive kernel KS(λ, ζ ) = (I −
S(λ)S(ζ )
∗
) · kd (λ, ζ ) is a natural multivariable generalization of the classical de Branges–Rovnyak canonical model space. A special
feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints M
∗
λj
of
the multiplication operators Mλj :f (λ) →λjf (λ) on H(kd ).We show that invariance of H(KS) under M
∗
λj
for each j = 1, . . . , d
is equivalent to the existence of a realization for S(λ) of the form S(λ) = D +C(I −λ1A1−· · ·−λdAd )
−1(λ1B1+· · ·+λdBd )
such that connecting operator U =
⎡
⎢⎣
A1 B1
...
...
Ad Bd
C D
⎤
⎥⎦
has adjoint U∗ which is isometric on a certain natural subspace (U is “weakly coisometric”)
and has the additional property that the state operators A1, . . . , Ad pairwise commute; in this case one can take the state
space to be the functional-model space H(KS) and the state operators A1, . . . , Ad to be given by Aj
=M
∗
λj
|H(KS ) (a de Branges–
Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where
the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of
such a realization such that the state operators A1, . . . , Ad satisfy an additional stability property.
© 2007 Elsevier Inc. All rights reserved.
