Let
D
D
be a Dirac type operator on a compact manifold
M
{\mathcal {M}}
and let
Σ
\Sigma
be a Lipschitz submanifold of codimension one partitioning
M
{\mathcal {M}}
into two Lipschitz domains
Ω
±
\Omega _{\pm }
. Also, let
H
±
p
(
Σ
,
D
)
{\mathcal {H}}^{p}_{\pm }(\Sigma ,D)
be the traces on
Σ
\Sigma
of the (
L
p
L^{p}
-style) Hardy spaces associated with
D
D
in
Ω
±
\Omega _{\pm }
. Then
(
H
−
p
(
Σ
,
D
)
,
H
+
p
(
Σ
,
D
)
)
({\mathcal {H}}^{p}_{-}(\Sigma ,D),{\mathcal {H}}^{p}_{+}(\Sigma ,D))
is a Fredholm pair of subspaces for
L
p
(
Σ
)
L^{p}(\Sigma )
(in Kato’s sense) whose index is the same as the index of the Dirac operator
D
D
considered on the whole manifold
M
{\mathcal {M}}
.