201
a as before, 0 :::; a < 1, .An = /-Ln, An = /-Ln+b where the /-Ln are the eigenvalues of A.
Hypotheses (1.2) to (1.14) are easily verified. For the exponential dichotomy we recall
that if Wm, m ;:: 0, are the eigenvectors of A (Aw m = /-Lmwm), and if Uo = 2.:::=1 UOmW m
then
00
-At
' \ ' -/tmt
e
Uo = ~ e
UOmW m ,
(3.9)
m=1
and (1.12), (1.13) hold with kl, k2 replaced by 1 in (1.12) and in the second and third
inequality (1.13). Of course Pn, the eigenprojector associated with /-LI, ... , /-Ln is orthogonal.
With a similar choice of spaces we can consider the case where E = H, A = Ao + AI,
Ao self-adjoint positive operator like the operator A before (eigenvectors Wm , eigenvalues
/-Lm, Aowm = /-Lmwm)' and Al is a skew-symmetric unbounded operator with domain
D(Ao) in H and such that
(3.10)
Then we set F = D(A'), E = D(A I+O!) as before and Pn is the eigenprojector of Ao
associated with the eigenvalues /-LI, ... , /-Ln. This is an orthogonal projector and, thanks
to (3.10), it is also an eigenprojector of A.
Another case of interest (see Section 4) is when E = H as before, A = Ao + b, Ao
is again self-adjoint positive unbounded in H, and b is a linear unbounded operator in
H dominated by Ao in the sense that D(Ag) C D(b) and bA-O! is bounded. Then we
can take F = H, E = D( Ag) and if /-Ln, Wn are the eigenvalues and eigenvectors of Ao,
.An = Iln + c' Il~,
An = Iln+! - C' /-L~+!,
for some c' > 0; see Section 4 for more details.
3.2.2. The hypotheses (2.1) and (2.2)
In all the cases in Section 3.2.1, .An = Iln, or .An ~ Iln as n -+ 00. Now, for large
classes of operators Ao defined by an elliptic boundary value problem, the behaviour of
J1.n for large n's is known.
Iln ~ cllon P , as n -+ 00,
(3.11)
for some c,p, J1.o > O. In fact (see S. Agmon (1965), R. Courant and D. Hilbert (1953) or
G. Metivier (1978) for the Stokes operator), we know for many elliptic operators that
p = 2r/m
(3.12)
where 2r is the order of the operator and m the space dimension. For some particular
geometrical domains, more precise informations on the eigenvalues are available.
a as before, 0 :::; a < 1, .An = /-Ln, An = /-Ln+b where the /-Ln are the eigenvalues of A.
Hypotheses (1.2) to (1.14) are easily verified. For the exponential dichotomy we recall
that if Wm, m ;:: 0, are the eigenvectors of A (Aw m = /-Lmwm), and if Uo = 2.:::=1 UOmW m
then
00
-At
' \ ' -/tmt
e
Uo = ~ e
UOmW m ,
(3.9)
m=1
and (1.12), (1.13) hold with kl, k2 replaced by 1 in (1.12) and in the second and third
inequality (1.13). Of course Pn, the eigenprojector associated with /-LI, ... , /-Ln is orthogonal.
With a similar choice of spaces we can consider the case where E = H, A = Ao + AI,
Ao self-adjoint positive operator like the operator A before (eigenvectors Wm , eigenvalues
/-Lm, Aowm = /-Lmwm)' and Al is a skew-symmetric unbounded operator with domain
D(Ao) in H and such that
(3.10)
Then we set F = D(A'), E = D(A I+O!) as before and Pn is the eigenprojector of Ao
associated with the eigenvalues /-LI, ... , /-Ln. This is an orthogonal projector and, thanks
to (3.10), it is also an eigenprojector of A.
Another case of interest (see Section 4) is when E = H as before, A = Ao + b, Ao
is again self-adjoint positive unbounded in H, and b is a linear unbounded operator in
H dominated by Ao in the sense that D(Ag) C D(b) and bA-O! is bounded. Then we
can take F = H, E = D( Ag) and if /-Ln, Wn are the eigenvalues and eigenvectors of Ao,
.An = Iln + c' Il~,
An = Iln+! - C' /-L~+!,
for some c' > 0; see Section 4 for more details.
3.2.2. The hypotheses (2.1) and (2.2)
In all the cases in Section 3.2.1, .An = Iln, or .An ~ Iln as n -+ 00. Now, for large
classes of operators Ao defined by an elliptic boundary value problem, the behaviour of
J1.n for large n's is known.
Iln ~ cllon P , as n -+ 00,
(3.11)
for some c,p, J1.o > O. In fact (see S. Agmon (1965), R. Courant and D. Hilbert (1953) or
G. Metivier (1978) for the Stokes operator), we know for many elliptic operators that
p = 2r/m
(3.12)
where 2r is the order of the operator and m the space dimension. For some particular
geometrical domains, more precise informations on the eigenvalues are available.