8 2. FUNCTIONAL MODELS

whole space /C. From the point of view of operator algebras, a row unitary c/-tuple

hi = (JA\,... ,Ud) amounts to a representation of the Cuntz algebra Od (see e.g.

[Da96] for definitions and further details). For any d-tuple hi — (Ui,... Md) °f op-

erators on a Hilbert space AC, it is convenient to introduce a functional calculus with

respect to the free semigroup Td on d letters # i , . . . , Q&. Namely, if v — gin .. . gix is

a word in Td (where i

n

, . . . , i\ £ { 1 , . . . , i}), we define hi to the power v, denoted

asW", by

The d-tuple W is defined as W = (Z^, ... Md)- We shall often work with products

of the form UWU*V. Note that if hi — (Ui-... Md) is row-unitary, then an expression

of the form U*vhiw collapses according to the following formula which will require

some additional explanation:

(2.2.1) WVUU

if \v\ \w\

if \w\ \w\.

Here \v\ refers to the length of the word v (\v\ = n if v = gin .. .g^), vT refers to

the transpose of v (vT = g^ .. . gin if v = gin .. . g^) and v~1 refers to the inverse

of v (v~l — g~\ .. g^ if v = gin .. . g^). However we do not interpret products of

the form v~1w in the free group generated by ( g i , . . . , #d); rather we define

_ i ( 0 ife = k

1 undefined if £ 7^ k

where 0 is the empty word of zero length equal to the unit element for Td For

an operator d-tuple, we interpret U® — I and ^ u n d e f i n e d = 0. This completes the

interpretation of the formula (2.2.1).

Suppose now that we are given a row-unitary d-tuple hi = (Ui,... Md) °

n a

Hilbert space /C together with some fixed subspace £ of /C. We may then associate

a matrix W = [lfV)W;Q^] with rows indexed by elements (v,w) of Td x Td and

columns indexed by elements (a, f3) of Td x Td given by

(2.2.2) WViW]ai/3 = PsUwU*vUa~YU^T \E.

It is straightforward to verify that this matrix has a Haplitz structure (see [BaV04]),

namely,

(2-2.3) Wft^-ag^p = WQ^g.^p,

(2-2-4) WVtW.,agi,p = Wvg-^w.atf3

as well as what we call the Cuntz property

d

(2-2.5) W0,™;0,/3 = ^2^w9j-AP9j-

3 = 1

Note that (2.2.3) is a Hankel-like property while (2.2.4) is a Toeplitz-like property

(and hence the term Haplitz). Property (2.2.5) is equivalent to the row unitary

property of hi (i.e., J2j^i^j^j ~ IK), i-e-? to hi inducing a representation of the

Cuntz algebra. Given that W = W^ (where [*] denotes the conjugate transpose