The cokernel of the trajectory operator #
Let A be an essentially bounded family of operators on a real inner product space E, and let
D_A u = u' + A u : W^{1,2}(ℝ, E) → L²(ℝ, E)
be its trajectory operator (TauCeti.WeakDerivStep.trajectoryOperator, for an L∞ class of A).
This file identifies the orthogonal complement of the range of D_A with the kernel of the formal
adjoint -d/ds + A†, where A† is the pointwise adjoint of A.
A path v orthogonal to the range pairs to zero with D_A (φ • e) for every test function φ and
every vector e. This says that v has weak derivative A† v, so v lies in W^{1,2}(ℝ, E) and
solves the formal adjoint equation -v' + A† v = 0: it is in the kernel of D_{-A†}. This
inclusion needs neither finite dimensionality of E nor any further hypothesis on A, and is
stated in that generality, for any essentially bounded B with B = A† almost everywhere
(TauCeti.WeakDerivStep.exists_ae_eq_adjoint provides one). Conversely, when E is finite
dimensional, density of test functions in W^{1,2}(ℝ, E) shows that every element of the kernel of
D_{-A†} is orthogonal to the range of D_A.
When the range of D_A is closed, its cokernel is the orthogonal complement of the range, so the
index of D_A is dim ker D_A - dim ker D_{-A†}. For symmetric A it reads
dim ker D_A - dim ker D_{-A}. The file TauCeti.Analysis.ODE.EnergyEstimate.Lyapunov uses this
to prove that D_A is Fredholm when the limits of A at ±∞ are hyperbolic.
Main declarations #
TauCeti.WeakDerivStep.hasWeakLineDerivOn_of_mem_orthogonal_range_trajectoryOperator: a path orthogonal to the range ofD_Ahas weak derivativeA† v.TauCeti.WeakDerivStep.orthogonal_range_trajectoryOperator_le_map_ker: such a path is the value of an element of the kernel ofD_{-A†}.TauCeti.WeakDerivStep.map_ker_le_orthogonal_range_trajectoryOperatorandTauCeti.WeakDerivStep.orthogonal_range_trajectoryOperator_eq_map_ker: the values of the kernel ofD_{-A†}are exactly the orthogonal complement of the range ofD_A.TauCeti.WeakDerivStep.index_trajectoryOperator_of_isClosed_range: whenever the range ofD_Ais closed, its index isdim ker D_A - dim ker D_{-A†}.TauCeti.WeakDerivStep.surjective_trajectoryOperator_iff_ker_neg_eq_bot: when the range ofD_Ais closed,D_Ais onto exactly when the kernel ofD_{-A†}vanishes.TauCeti.WeakDerivStep.surjective_trajectoryOperator: for aC¹symmetric family with a uniform spectral gapαand|⟪v, A' v⟫| ≤ β ‖v‖²,β < α², the operatorD_Ais onto.TauCeti.WeakDerivStep.exists_ae_eq_adjoint: the pointwise adjoint of an essentially bounded family has anL∞class.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.1 (the cokernel of a first-order operator through its formal adjoint).
- M. Schwarz, Morse Homology, Chapter 2 (the Fredholm property of linearized trajectory operators).
A path orthogonal to the range solves the adjoint equation weakly. Let B be almost
everywhere the pointwise adjoint of the coefficient of the trajectory operator D_A u = u' + A u.
Then every v ∈ L²(ℝ, E) orthogonal to the range of D_A has weak derivative B v: it is a weak
solution of the formal adjoint equation -v' + B v = 0.
Paths orthogonal to the range lie in the kernel of the formal adjoint. Let B be almost
everywhere the pointwise adjoint of the coefficient of D_A u = u' + A u. Every v ∈ L²(ℝ, E)
orthogonal to the range of D_A is the value of an element of W^{1,2}(ℝ, E) in the kernel of
D_{-B}; that is, v solves v' - B v = 0 in W^{1,2}.
The kernel of the formal adjoint is orthogonal to the range. Let B be almost everywhere
the pointwise adjoint of the coefficient of D_A u = u' + A u, and let E be finite dimensional.
The value of every solution of v' - B v = 0 in W^{1,2}(ℝ, E) is orthogonal to the range of
D_A.
The orthogonal complement of the range of D_A is the kernel of its formal adjoint. Let
B be almost everywhere the pointwise adjoint of the coefficient of D_A u = u' + A u, and let
E be finite dimensional. The orthogonal complement of the range of D_A in L²(ℝ, E) consists
exactly of the values of the solutions of v' - B v = 0 in W^{1,2}(ℝ, E), the kernel of
D_{-B}.
The index of a trajectory operator with closed range. Let B be almost everywhere the
pointwise adjoint of the coefficient of D_A u = u' + A u, let E be finite dimensional and let
the range of D_A be closed. Then the index of D_A is dim ker D_A - dim ker D_{-B}: its
cokernel is the kernel of the formal adjoint -d/ds + B, whose solutions are those of
v' - B v = 0.
A trajectory operator with closed range is onto exactly when its formal adjoint is
injective. Let B be almost everywhere the pointwise adjoint of the coefficient of
D_A u = u' + A u, let E be finite dimensional and let the range of D_A be closed. Then D_A
is surjective if and only if the only solution of v' - B v = 0 in W^{1,2}(ℝ, E) is zero, that
is, the kernel of D_{-B} vanishes.
An L∞ class of the pointwise adjoint. If AL is an L∞ class of A, then the pointwise
adjoints s ↦ A(s)† also have an L∞ class. It is the coefficient of the formal adjoint
-d/ds + A† of the trajectory operator d/ds + A.
An L∞ class of the pointwise adjoint of A is almost everywhere the pointwise adjoint of an
L∞ class of A. This is the hypothesis on the coefficient of the formal adjoint in
TauCeti.WeakDerivStep.orthogonal_range_trajectoryOperator_eq_map_ker.
The trajectory operator of a coercive symmetric family is onto. If a C¹ family of
symmetric operators on a finite-dimensional space satisfies α ‖v‖ ≤ ‖A s v‖ and
|⟪v, A' s v⟫| ≤ β ‖v‖² everywhere with β < α², then D_A u = u' + A u maps W^{1,2}(ℝ, E)
onto L²(ℝ, E). Together with TauCeti.WeakDerivStep.ker_trajectoryOperator_eq_bot, D_A is
then a linear isomorphism, with the inverse bound of
TauCeti.WeakDerivStep.min_one_mul_norm_le_norm_trajectoryOperator. The coefficient of the
operator is any L∞ class AL of A.