12 2. FUNCTIONAL MODELS
where we have set kper(zX) equal to the "perverse Szego kernel"
(2.2.26) fc^CH^-Y'^Cy
and where z' = (z[,..., z^) is another set of noncommuting indeterminants, each
of which commutes with z±,... ,Zd, Cii •• -Cd
(see
Proposition 2.7 in [BaV04]).
Here it is understood that one does the evaluation in z' followed by the evaluation
z' z before performing the multiplication by kper(zX)- For the case where
p(z) =
^2aPaZa
is an analytic polynomial in V{Td x {0},£), then
(2.2.27) W\p](z, C) = W(0, Okper(z, C)p(z) + W(z, ()p(z).
If in addition W(0,() =
I(0,
then (2.2.27) simplifies further to
(2.2.28) W\p](z, 0 = W(z, C)p(z) if W(0, () = I(®.
The symbol of the adjoint W^
0
f W is given by
(2.2.29) W^](zX) = W((,zy.
(where we use the convention that
(zvY
=
zv
and (C™)* =
(w
)- Thus the
selfadjointness of a [*]-Haplitz operator W can be expressed directly in terms of
the symbol: W = W^ (as a [*]-Haplitz operator) if and only if
(2.2.30) W(z,() = W((,zy
Furthermore, a selfadjoint Haplitz operator is positive semidefinite if and only if its
symbol W(z, £) and the Cuntz defect D^(z, £) of its symbol given by
d
Dw(z,C) = W(z,Q ~ Y.^WiZ'CKk1-
fc=l
have factorizations
(2.2.31) W(z,C) = Y(C)Y(zy,
d
(2.2.32) w(z, Q-J2 ^(z,
0C71
= r(C)rw*
3 = 1
Here Y(z) is a formal series of the form Y(z) ^2we:F Ywzw where each Yw is
an operator from some auxiliary Hilbert space Ji to £, and similarly, T{z) is a
formal series of the form T(z) = ^2we:F
Fwzw.
Furthermore, the Haplitz operator
W is a Cuntz weight if and only if its symbol W(zX) is a positive symbol (i.e., a
factorization (2.2.31) exists) and its Cuntz defect D^(zX) is zero:
d
(2.2.33) W(z, ()-J2
^lW{z, OC"1
= 0.
Moreover, the Haplitz operator W = \Wv,W]OL^\ with matrix entries
(2.2.34) WVtW.iaip =
P£UwWvUaTU^T
\E
(or, equivalently, its symbol W(zX)) is a complete unitary invariant for a row-
unitary U together with a specified choice of *-cyclic subspace £. For complete
details, we refer to [BaV04].
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