Erratum to: Generalizing the Kantorovich Metric to Projection Valued Measures
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Erratum to: Generalizing the Kantorovich Metric to Projection Valued Measures Trubee Davison1
© Springer Science+Business Media Dordrecht 2015
Erratum to: Acta Appl. Math. (2014) DOI 10.1007/s10440-014-9976-y There is an error in the proof of Theorem 2.15 in the article Generalizing the Kantorovich Metric to Projection Valued Measures [1]. The result is still true. The theorem is restated below: Theorem 2.15 [1] The map Φ : P (X) → P (X) given by E(·) →
N−1
Si E σi−1 (·) Si∗
i=0
is a Lipschitz contraction in the ρ metric. The following is a corrected version of the part of the proof which contains the error. The author wants to acknowledge Krystal Taylor (University of Minnesota) for identifying the error. The following argument replaces the argument which begins on the line after Claim 2.16. Let E, F ∈ P (X). Recall that r = max0≤i≤N−1 {ri }, where ri is the Lipschitz constant associated to σi , and note that 0 < r < 1. Choose φ ∈ Lip1 (X), and h ∈ H with h = 1. Then
φdΦ(E) − φdΦ(F ) h, h The online version of the original article can be found under doi:10.1007/s10440-014-9976-y.
B T. Davison
[email protected]
1
Department of Mathematics, University of Colorado, Boulder, USA
T. Davison
φdΦ(F ) h, h = φdΦ(E)h,h − φdΦ(F )h,h = φdΦ(E) h, h − X
N−1 −1 N−1 −1 φdESi∗ h,Si∗ h σi (·) − φdFSi∗ h,Si∗ h σi (·) = i=0 X i=0 X N−1 N−1 ∗ ∗ ∗ ∗ (φ ◦ σi )dESi h,Si h − (φ ◦ σi )dFSi h,Si h = i=0 X i=0 X N−1 (φ ◦ σi )dESi∗ h,Si∗ h − (φ ◦ σi )dFSi∗ h,Si∗ h = X X i=0 N−1 φ ◦ σi φ ◦ σi r = dESi∗ h,Si∗ h − dFSi∗ h,Si∗ h r r X X
X
i=0
N−1 φ ◦ σ φ ◦ σ i i ≤r dESi∗ h,Si∗ h − dFSi∗ h,Si∗ h r r X i=0 X N−1
φ ◦ σ φ ◦ σ i i ∗ ∗ dE − dF Si h, Si h =r r r i=0 N−1
∗ 2 φ ◦ σi φ ◦ σi
. ≤r dE − dF Si h
r r i=0 Note that the function
φ◦σi r
∈ Lip1 (X) for all 0 ≤ i ≤ N − 1. Hence
N−1
∗ 2 φ ◦ σ φ ◦ σ i i
r dE − dF
Si h r r i=0 N−1 N−1 ∗ ∗ ∗ ≤ rρ(E, F ) Si h, Si h = rρ(E, F ) Si Si h, h i=0
i=0
N−1 ∗ = rρ(E, F ) Si Si h, h = rρ(E, F ) h, h = rρ(E, F ). i=0
Therefore
φdΦ(E) − φdΦ(F ) ≤ rρ(E, F ).
Since φ is an arbitrary element of Lip1 (X), ρ Φ(E), Φ(F ) ≤ rρ(E, F ). This proves that Φ is a Lipschitz contraction in the ρ metric on P (X).
Erratum to: Generalizing the Kantorovich Metric to Projection Valued Measures
References 1. Davison, T.: Generalizing the Kantorovich metric to projection-valued measures. Acta Appl. Math. (2014). doi:10.1007/s10440-014-9976-y
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