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46 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ
we can write, with Φ = ⊕∞ j=0Φj ∈ ⊕∞ j=0Hj+ = F+,
〈Φ,AΦ〉 = 〈fMΦ,AfMΦ〉+〈gMΦ,AgMΦ〉
+ � �
(fM (i)−fM (j)) 2 +(gM (i)−gM (j)) 2�
〈Φi,AΦj〉.
1≤|i−j|≤σ
Since f and g are smooth, we have
(fM (i)−fM (j)) 2 +(gM (i)−gM (j)) 2 ≤ � ||f ′ ||∞ +||g ′ � (i−j)
||∞
2
M2 .
Moreover, using the assumption that A ≥ 0 we get
�
1≤|i−j|≤σ
|〈Φi,AΦj〉| ≤ �
1≤|i−j|≤σ
(〈Φi,AΦi〉+〈Φj,AΦj〉)
≤ 4σ �
〈Φi,AΦi〉 = 4σ〈Φ,A0Φ〉.
Therefore, we obtain the operator inequality (54).
Finally, let us show that if
δ := sup{||gMΦ|| 2 : Φ ∈ Y,||Φ|| = 1} < (dimY) −1 ,
then dim(fMY) = dimY. In fact, assume that dimY = L and let an
orthonormal basis {Φi} L i=1 for Y. For all {αi} L i=1 ∈ CL \{0}, we have
�
� L� �
�
�
=
i=1
≥
L�
i=1
αifMΦi
�
�
�
�
�
2
≥
L�
|αi| 2 �fMΦi� 2 −2 �
i=1
|αi| 2�
1−�gMΦi� 2�
−2 �
L�
|αi| 2 (1−δ)− �
i=1
1≤i