Erratum to “The Partial Inner Product Space Method: A Quick Overview”

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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 𝑉_𝐼,𝑌_𝐾 be two LHSs or LBSs. An operator 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) is called a homomorphism if(i)for every 𝑟∈𝐼, there exists 𝑢∈𝐾 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist;(ii)for every 𝑢∈𝐾, there exists 𝑟∈𝐼 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist.

Equivalently, for every 𝑟∈𝐼, there exists 𝑢∈𝐾 such that (𝑟,𝑢)∈𝑗(𝐴) and (𝑟,𝑢)∈𝑗(𝐴), and for every 𝑢∈𝐾, there exists 𝑟∈𝐼 with the same property.

The definition may be rephrased as follows: 𝐴∶𝑉_𝐼→𝑌_𝐾 is a homomorphism ifpr₁𝗃(𝐴)∩𝗃(𝐴)=𝐼,pr₂𝗃(𝐴)∩𝗃(𝐴)=𝐾, (1) where 𝗃(𝐴)={(𝑟,𝑢)∶(𝑟,𝑢)∈𝗃(𝐴)} 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 𝗃(𝐴)=𝐼×𝐾 and 𝗃(𝐴^×)=𝐾×𝐼.

We denote by Hom(𝑉_𝐼,𝑌_𝐾) the set of all homomorphisms from 𝑉_𝐼 into 𝑌_𝐾. The following property is easy to prove:

Let 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾). Then, 𝑓#_𝐼𝑔 implies 𝐴𝑓#_𝐾𝐴𝑔.