Proposition
(Boundedness of surjection) In context
of the condition (
Space splitting setup
)
and for a stable splitting, the operator
is bounded and
Proof
For any
we
estimate
Proposition
(Boundedness of adjoint of
surjection) In context of the condition
(
Space splitting setup
) and for a stable
splitting, the operator
is bounded and
Proof
The symmetry follows from the proposition
(
Form of Schwarz operator
) and
generic properties of taking
adjoint:
For any
we introduce a class
.
The class
is never empty by surjectivity of
,
see the condition (
Space splitting
setup
). Then, for any
and
we
calculate
where the equality takes place for
and
.
We
obtained
or
with equality reached at
.
Thus
Next, we establish the upper bound for
.
We calculate, using the stability of splitting and
,
thus
or
Next, we establish the lower bound for
.
We use again the surjectivity of
,
see the condition (
Space splitting
setup
), to introduce a non empty class
.
Then, for any
and any
At this point we take infimum with respect to
and use the definition of
,
We use stability of
splitting,
Thus
or
Therefore,
is invertible. The formula
now follows from
.
