2.2. FORMAL POWE R SERIES
7
where Uw
:
Cw ~ Cw is the operator of multiplication by the indeterminant z on
Cw'
U\v- f(z) «- 2/(2),
and we recover W as PF = $1M (where $[*' is the formal adjoint of $ with
respect to the /C-inner product on its domain and the formal L2-inner product on
its range). In particular, Uw is unitary on Cw-, and, if £ is *-cyclic for W, then $
implements a unitary equivalence between U on /C and the functional model ZYw
on Cw- Conversely, if W is any positive-semidefinite block-Toeplitz matrix, we
may define a space Cw as above, and an operator Uw '•
f{z)
^
zf(z) o n
£w- The
Toeplitz structure of W guarantees that Cw is unitary. It is often more convenient
to work with the symbol of W, the formal Laurent series W(z) = X]^L-oo Wnzn
with operator coefficients determined by W(z) e = (We)(z) for e G £ (considered
as a constant function in P(Z, £)). Equivalently, we have Wij Wi-j. If one
interprets the formal indeterminate z as a variable inside the unit disk D, the
formula W(z) = X^^L-oo
Wnzn
defines a harmonic function on the unit disk which
therefore has a Poisson representation
^'H0^MT)21
for a positive operator-valued measure r on T. Some authors prefer to work with
the analytic function with positive-real part on the unit disk having the Herglotz
representation
f0
27r
Jr
T
-
z
Closely related functional models for unitary operators have appeared in the lit-
erature (see [BoDKOO] and [BraR66]). Indeed, the space Cw can be identi-
fied as the space of all formal power series f(z) = ]Z^L-oo fnZn such that fn
-^
fTT~n
dv(r) where the vector measure v (called a chart in the terminology of
[BoDKOO]) is in the Hellinger space
Ca
delineated in [BoDKOO]. Alternatively,
f(z) = S^L-oo fnzU ^ £w if and only if the pair of analytic functions (f(z)^(z))
defined by f(z) = Yl™=o fnzU and g(z) = J2^Lo f-n-izU for z in the unit disk D
is in the space £((f) introduced in [BraR66].
2.2. Functional models of noncommutative formal power series
The paper [BaV04] presents an extension of these ideas to the setting where
the single unitary operator U is replaced by a row-unitary operator-tuple U =
(U\,... Md)- Given a d-tuple of operators U = (Wi,... Md) on a Hilbert space /C,
in general we say that U is a row isometry (respectively, row unitary) if the operator
matrix
r/ci
[Wi ... Ud] :
|_/CJ
is isometric (respectively, unitary) as an operator from ©jf=1/C to /C. More con-
cretely, U is row-isometric if each of Uj is an isometry, and the image spaces im Uj
are pairwise orthogonal for j =
1,...,G L
The d-tuple U is row-unitary if U is
a row-isometry such that the span of the images imUj over j = 1,... , d is the
/C
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