Erratum to “The Partial Inner Product Space Method: A Quick Overview”
First published: 11 May 2011
The definition of homomorphism given in Section 5.2.2 is incorrect. Here is the exact definition. The rest of the discussion is correct.
Let VI, YK be two LHSs or LBSs. An operator A ∈ Op(VI, YK) is called a homomorphism if
- (i)
for every r ∈ I, there exists u ∈ K such that both Aur and exist;
- (ii)
for every u ∈ K, there exists r ∈ I such that both Aur and exist.
Equivalently, for every r ∈ I, there exists u ∈ K such that (r, u) ∈ j(A) and , and for every u ∈ K, there exists r ∈ I with the same property.
The definition may be rephrased as follows: A : VI → YK is a homomorphism if
(1)
where and pr1, pr2 denote the projection on the first, respectively, the second component.Contrary to what is stated in [1, Definition 3.3.4], the condition (1), which is the correct one, does not imply and .
We denote by Hom(VI, YK) the set of all homomorphisms from VI into YK. The following property is easy to prove:
Let A ∈ Hom(VI, YK). Then, f#Ig implies Af#KAg.
