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Chapter 5. Extension Bergman distance
Now we take the square of their sum and use the Cauchy-Schwarz
inequality in the second step.
(d (A n d , B n d ) + d (B n d , C n d )) 2 �x0� 2
≥
≥
≥
≥
∞� 1
∞� 1
k
k=1
�(Akd − B k d )x0� 2 +
k
k=1
�(Bk d − C k d )x0� 2
�
�
�
+ 2�
∞ �
∞�
k=1
k=1
1
k �(Ak d − Bk d
1
k �(Ak d − B k d )x0� 2 +
+ 2
∞�
k=1
∞�
k=1
∞�
k=1
�
�
�
)x0�2� ∞ �
∞�
k=1
k=1
1
k �(Bk d − Ck d )x0�2 1
k �(Bk d − C k d )x0� 2
1
k �(Ak d − B k d )x0��(B k d − C k d )x0�
1 � k
�(Ad − B
k
k d )x0� + �(B k d − C k d )x0� �2
1
k �(Ak d − C k d )x0� 2 , (5.4)
where we used the triangle inequality for the norm of X in the last
step. Since inequality (5.4) holds for all x0 ∈ X, we have
d � e At , e Ct�2 ≤ � d � e At , e Bt � + d � e Bt , e Ct��2 .
Hence the triangle inequality holds.
The Bergman distance defines a metric. �
5.2 Class properties
In this section we study the elements belonging to one equivalence class.
First we show that if A generates a C0-group then the other elements in its
class also generate a C0-group. Furthermore, we define stable states and
show that these stable state are the same for the generators in an equivalence
class.
Lemma 5.5 Let A and à generate a C0-semigroup and A B ∼ Ã. If A
generates a C0-group on X, then also à generates a C0-group on X.
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