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Chapter 5. Extension Bergman distance
where we used the triangle inequality for the norm of X in the last
step. Since inequality (5.2) 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. �
5.1.2 Classes of cogenerators
The Bergman distance on the Hilbert space X can be seen as a binary
B
relation on the set of cogenerators on X. We define Ad ∼ Ãd if the cogenerators
A and à have a finite Bergman distance. that is if the following two
equations hold.
∞�
k=1
∞�
k=1
1
k �(Ak d − Ãk d)x0� 2 < ∞,
1
k �(A∗k
d − Ã∗k
d )x0� 2 < ∞.
These equations are the same as equations (4.5) and (4.6).
Lemma 5.3 The binary relation B ∼ on the set of cogenerators on X, is an
equivalence relation.
Proof: We have to check the reflexivity, symmetry and transitivity of B ∼.
B
B
The reflexivity, Ad ∼ Ad, and the symmetry, if Ad ∼ Ãd then Ãd
B
∼ Ad, are
trivial. So it remains to show transitivity. Using the inequality
the transitivity, if Ad
∞�
k=1
1
k �(Ak d − C k d )x0� 2 ≤
= 2
∞�
k=1
�x − z� 2 ≤ 2�x − y� 2 + 2�y − z� 2 ,
B
B
B
∼ Bd and Bd ∼ Cd then Ad ∼ Cd, is easy to see from
∞�
k=1
1 � k
2�(Ad − B
k
k d )x0� 2 + 2�(B k d − C k d )x0� 2�
1
k �(Ak d − B k d )x0� 2 + 2
∞�
k=1
1
k �(Bk d − C k d )x0� 2 < ∞.
And the same holds for the adjoint operators Ãd, ˜ Bd, and ˜ Cd.
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