206
We assume also that
(4.11)
and
p(1 - a) > 1.
(4.12)
As we shall see in the examples, hypothesis (4.12) related to the spectral gap condition
will be the most restrictive one.
The operator b is a linear unbounded operator in H with domain D(b)j it is dominated by Ao in the following sense
D(A~) C D(b), for some a, 0:5 a < 1,
(4.13)
b Ail" is bounded in H.
(4.14)
Of course, because of (4.13), D(A) = D(Ao).
Concerning the eigenvectors and eigenvalues of A, we have, thanks to (4.13) and
(4.14), the following important properties which we state without proof:
(i) The spectrum of A = Ao +b consists of eigenvalues of finite multiplicity. Moreover if
we denote by Vj j ~ 1, these eigenvalues arranged so that the sequence {Re Vj} jEN*
is nondecreasing, then
Re(Vj) '" ).,j, as j --+ 00.
(ii) There exists a constant Co > 0 such that
u(A) C U B().,j,co).,j),
jEN*
(4.15)
(4.16)
where u(A) is the spectrum of A and B().,j,co).,'f) the ball of C centered at ).,j of
radius co).,j. Moreover if)., ¢ UjEN B().,j,co).,'f),
II(A - ).,)-111 C(H) :5 211(Ao - ).,)-IIIC(H) ,
(4.17)
II(A - ).,)-1 - (Ao - ).,)-1 II "(H) :5 Co sup I/,i I sUP-I' 1 I'
(4.18)
'"
i>O A -!li i>O A - !li
Let {Vj}jEN denote a system of generalized eigenvectors (rootvectors) of A
associated with the IIi'S:
(4.19)
(iii) We have
The system {Vj} jEN* is total in H.
(4.20)