Suppose that
ϕ
=
ψ
z
γ
\phi =\psi z^\gamma
where
γ
∈
Z
+
\gamma \in Z_+
and
ψ
∈
Lip
β
,
1
2
>
β
>
1
\psi \in \operatorname {Lip}_\beta ,\,{1\over 2}>\beta >1
, and the Toeplitz operator
T
ψ
T_\psi
is invertible. Let
D
n
(
T
ϕ
)
D_n(T_\phi )
be the determinant of the Toeplitz matrix
(
(
ϕ
^
i
,
j
)
)
=
(
(
ϕ
^
i
−
j
)
)
,
0
≤
i
,
j
≤
n
,
((\hat \phi _{i,j}))=((\hat \phi _{i-j})),\quad 0\leq i,j\leq n ,
where
ϕ
^
k
=
1
2
π
∫
0
2
π
ϕ
(
θ
)
e
−
i
k
θ
d
θ
\hat \phi _k={1\over 2\pi }\int _0^{2\pi } \phi (\theta )e^{-ik\theta }\, d\theta
. Let
P
n
P_n
be the orthogonal projection onto
ker
S
∗
n
+
1
=
⋁
{
1
,
e
i
θ
,
e
2
i
θ
,
…
,
e
i
n
θ
}
,
\ker {S^*}^{n+1}=\bigvee \{1,e^{i\theta }, e^{2i\theta },\ldots , e^{in\theta }\},
where
S
=
T
z
S=T_z
; set
Q
n
=
1
−
P
n
Q_n=1-P_n
, let
H
ω
H_\omega
denote the Hankel operator associated to
ω
\omega
, and set
ω
~
(
t
)
=
ω
(
1
t
)
\tilde \omega (t)=\omega ({1\over t})
for
t
∈
T
t\in \mathbb {T}
. For the Wiener-Hopf factorization
ψ
=
f
g
¯
\psi =f\bar g
where
f
,
g
f, g
and
1
f
,
1
g
∈
Lip
β
∩
H
∞
(
T
)
,
1
2
>
β
>
1
{1\over f },{1\over g}\in \operatorname {Lip}_\beta \cap H^\infty (\mathbb {T}), {1\over 2}>\beta >1
, put
E
(
ψ
)
=
exp
∑
k
=
1
∞
k
(
log
f
)
k
(
log
g
¯
)
−
k
E(\psi )=\exp \sum _{k=1}^\infty k(\log f)_k(\log \bar g)_{-k}
,
G
(
ψ
)
=
exp
(
log
ψ
)
0
.
G(\psi )=\exp (\log \psi )_0.
Theorem A.
D
n
(
T
ϕ
)
=
(
−
1
)
(
n
+
1
)
γ
G
(
ψ
)
n
+
1
E
(
ψ
)
G
(
g
¯
f
)
γ
⋅
det
(
(
T
f
g
¯
z
n
+
1
⋅
[
1
−
H
g
¯
f
Q
n
−
γ
H
(
f
g
¯
)
~
]
−
1
z
α
−
1
,
z
τ
−
1
)
)
γ
×
γ
⋅
[
1
+
O
(
n
1
−
2
β
)
]
.
D_n(T_\phi )=(-1)^{(n+1)\gamma } G(\psi )^{n+1}E(\psi ) G({\bar g\over f})^\gamma \cdot \det \bigg ((T_{{f\over \bar g}z^{n+1}}\cdot [1-H_{\bar g\over f} Q_{n-\gamma } H_{({\frac {f}{\bar g}})^{\tilde {}}} ]^{-1}z^{\alpha -1},z^{\tau -1})\bigg )_{\gamma \times \gamma } \cdot [1+O(n^{1-2\beta })].
Let
H
2
(
T
)
=
X
∔
Y
H^2(\mathbb {T})= {\mathcal X}\dotplus {\mathcal Y}
be a decomposition into
T
ϕ
T
ϕ
−
1
T_\phi T_{\phi ^{-1}}
invariant subspaces,
X
=
⋂
n
=
1
∞
ran
(
T
ϕ
T
ϕ
−
1
)
n
{\mathcal X}= \bigcap _{n=1}^\infty \operatorname {ran} (T_\phi T_{\phi ^{-1}})^n
and
Y
=
⋃
n
=
1
∞
ker
(
T
ϕ
T
ϕ
−
1
)
n
{\mathcal Y}=\bigcup _{n=1}^\infty \ker (T_\phi T_{\phi ^{-1}})^n
, so that
T
ϕ
T
ϕ
−
1
T_\phi T_{\phi ^{-1}}
restricted to
X
{\mathcal X}
is invertible,
Y
{\mathcal Y}
is finite dimensional, and
T
ϕ
T
ϕ
−
1
T_\phi T_{\phi ^{-1}}
restricted to
Y
{\mathcal Y}
is nilpotent. Let
{
w
α
}
1
γ
\{w_\alpha \}_1^\gamma
be the basis
{
T
f
z
α
}
0
γ
−
1
\{T_f z^\alpha \}_0^{\gamma -1}
for the null space of
T
ϕ
T
ϕ
−
1
T_\phi T_{\phi ^{-1}}
, and let
u
α
u_\alpha
be the top vector in a Jordan root vector chain of length
m
α
+
1
m_\alpha +1
lying over
(
−
1
)
m
α
w
α
(-1)^{m_\alpha }w_\alpha
, i.e.,
(
T
ϕ
T
ϕ
−
1
)
m
α
u
α
=
(
−
1
)
m
α
w
α
(T_\phi T_{\phi ^{-1}})^{m_\alpha }u_\alpha =(-1)^{m_\alpha }w_\alpha
where
m
α
=
max
{
m
∈
Z
+
:
∃
x
so that
(
T
ϕ
T
ϕ
−
1
)
m
x
=
w
α
}
−
1
m_\alpha =\max \{m\in Z_+:\exists x\,\text {so that} (T_\phi T_{\phi ^{-1}})^mx=w_\alpha \}^{-1}
.
Theorem B.
E
(
ψ
)
G
(
g
¯
f
)
γ
=
E( \psi ) G({\bar g\over f})^\gamma =
∏
λ
∈
σ
(
T
ϕ
T
ϕ
−
1
)
∖
{
0
}
λ
det
(
u
α
,
T
1
g
z
τ
−
1
)
{\prod _{\lambda \in \sigma (T_{\phi } T_{\phi ^{-1}})\setminus \{0\}}\,\lambda }\over \det ( u_\alpha ,T_{1\over g}z^{\tau -1})
=
(
g
¯
∪
f
×
g
¯
f
∪
z
γ
)
(
T
)
=\left (\bar g\cup f\times {\bar g\over f}\cup z^\gamma \right )(\mathbb {T})
, the holonomy of a Deligne bundle with connection defined by the factorization
ϕ
=
f
g
¯
z
γ
\phi = f\bar gz^\gamma
.
Note that the generalizations of the Szegö limit theorem for
D
n
(
T
ϕ
)
D_n(T_\phi )
which have appeared in the literature with
1
1
instead of
[
1
−
H
g
¯
f
Q
n
−
γ
H
(
f
g
¯
)
~
]
−
1
[1-H_{\bar g\over f} Q_{n-\gamma } H_{({f\over \bar g})^{\tilde {}}}]^{-1}
have the defect that the limit of
D
n
(
T
ϕ
)
(
−
1
)
(
n
+
1
)
γ
G
(
ψ
)
n
+
1
det
(
T
f
g
¯
z
n
+
1
z
α
−
1
,
z
τ
−
1
)
{D_n(T_\phi )\over (-1)^{(n+1)\gamma } G(\psi )^{n+1} \det (T_{{f\over \bar g}z^{n+1}}z^{\alpha -1},z^{\tau -1})}
does not exist in general. An example is given with
D
n
(
T
ϕ
)
≠
0
D_n(T_\phi )\neq 0
yet
D
γ
−
1
(
T
f
g
¯
z
n
+
1
)
=
0
D_{\gamma -1}(T_{{f\over \bar g}z^{n+1}})=0
for infinitely many
n
n
.