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Corrigendum to “Continuity and generators of dynamical semigroups for infinite Bose systems” [J. Funct. Anal. 256 (5) (2009) 1453–1475]
Journal of Functional Analysis 259 (2010) 2455–2456 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Continuity and generators of dynamical semigroups for infinite Bose systems” [J. Funct. Anal. 256 (5) (2009) 1453–1475] Philippe Blanchard a , Mario Hellmich a,∗ , Piotr Ługiewicz b , Robert Olkiewicz b a Faculty of Physics and BiBoS, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany b Institute of Theoretical Physics, University of Wrocław, Pl. M. Borna 9, 50204 Wrocław, Poland
Received 21 June 2010; accepted 12 July 2010
After publication of our paper we discovered an error in the proof of Theorem 4, which holds only under an additional assumption. In fact we have the following. 0 ∗ ∗ Theorem 4. Suppose t → Tt (x) is weak -measurable for all x ∈ X0 . Then {Tt }t0 is weak continuous if and only if t>0 ker Tt = {0}.
Proof. If 0 = x ∈ t>0 ker Tt then there is ϕ ∈ X∗ with x, ϕ = 1, thus 0 = limt↓0 Tt (x), ϕ = x, ϕ = 1, so we obtain a contradiction. Conversely, consider the predual semigroup Tt,∗ : X∗ → X∗ . Since (ran Tt,∗ )⊥ = ker Tt we get ( t>0 ran Tt,∗ )⊥ = t>0 ker Tt = {0}, hence by the bipolar theorem t>0 ran Tt,∗ is weak and thus norm dense in X∗ . Next, since ball X is metrizable in the weak∗ -topology for any x ∈ ball X there is a sequence xn0 ∈ ball X0 with xn0 → x, so t → Tt (x) is weak∗ -measurable and t → Tt,∗ (ϕ) is weakly measurable. By the Pettis theorem we conclude that this map is strongly measurable. Finally we apply the Hille theorem about the strong continuity of semigroups. 2 Since Theorem 4 only holds under an additional assumption we have to modify the proof of Proposition 8. We strengthen the assumptions E1–E6 as follows: Let S be a complex Hilbert space with norm · ; we consider it as a symplectic space via σ (f, g) = Imf, g as well as a real Hilbert space (SR , ·,·R ) with f, gR = Ref, g. DOI of original article: 10.1016/j.jfa.2008.05.013. * Corresponding author.
E-mail address: [email protected] (M. Hellmich). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.012
2456
P. Blanchard et al. / Journal of Functional Analysis 259 (2010) 2455–2456
E1 The family of maps f → (F μt )(f ), t 0, is equicontinuous with respect to the nuclear topology τn (SR , · ). I.e. for any > 0 there exists δ > 0 and p ∈ Pn (S, · ) such that |1 − (F μt )(f )| < for any t 0 provided p(f ) < δ. E2 t → (F μt )(f ) is continuous for any f ∈ S. E3 The map S f → Wω (f ) is continuous when S is endowed with the norm topology and M with the strong operator topology. E4–E6 remain unchanged. ∗ group of automorProof of Proposition 8. Since {βt }t0 extends to a weak -continuous phisms on M it remains to consider Vt (x) = S Wω (g)xW (g)∗ dνt (g). By E3 the map g → Wω (g)xWω (g)∗ is weak∗ -continuous for all x ∈ M. The quasimeasure νt is σ -additive by E1 and thus extends to a measure. Now we have f, gR = −σ (if, g), thus eif,gR dνt (g) − 1 = eiσ (−if,g) dνt (g) − 1 = (F μt )(−if ) − 1 < SR
S
provided p(f ) = p(−if ) < δ. Hence the Fourier transforms νˆ t are equicontinuous at 0 with respect to the τn (SR , · )-topology on (SR , ·,·R ). Now by Theorem VI.2.3. of [39] the set {νt }t0 is relatively weakly compact in M1+ (S). By E2 the map t → νˆ t (f ) is continuous for any f ∈ SR , so we obtain that t → νt is weakly continuous by a standard result, i.e. t → ψ dνt is continuous for any bounded · -continuous function ψ . It follows that Vt (x) → x relative to the weak∗ -topology as t ↓ 0 for any x ∈ M. We conclude that t>0 ker Vt = {0}, so Tt = βt ◦ Vt is a weak∗ -continuous semigroup on M by Theorem 4. 2 We remark that it can be shown that with the original assumptions E1–E6 in the paper the conclusion of Proposition 8 holds if we additionally assume that limt↓0 S cω (g + g ) dνt (g) = cω (g ) for any g ∈ S, where cω is the generating functional of ω. With these changes all further results in the paper remain true. References [39] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
Corrigendum
Corrigendum to “Continuity and generators of dynamical semigroups for infinite Bose systems” [J. Funct. Anal. 256 (5) (2009) 1453–1475] Philippe Blanchard a , Mario Hellmich a,∗ , Piotr Ługiewicz b , Robert Olkiewicz b a Faculty of Physics and BiBoS, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany b Institute of Theoretical Physics, University of Wrocław, Pl. M. Borna 9, 50204 Wrocław, Poland
Received 21 June 2010; accepted 12 July 2010
After publication of our paper we discovered an error in the proof of Theorem 4, which holds only under an additional assumption. In fact we have the following. 0 ∗ ∗ Theorem 4. Suppose t → Tt (x) is weak -measurable for all x ∈ X0 . Then {Tt }t0 is weak continuous if and only if t>0 ker Tt = {0}.
Proof. If 0 = x ∈ t>0 ker Tt then there is ϕ ∈ X∗ with x, ϕ = 1, thus 0 = limt↓0 Tt (x), ϕ = x, ϕ = 1, so we obtain a contradiction. Conversely, consider the predual semigroup Tt,∗ : X∗ → X∗ . Since (ran Tt,∗ )⊥ = ker Tt we get ( t>0 ran Tt,∗ )⊥ = t>0 ker Tt = {0}, hence by the bipolar theorem t>0 ran Tt,∗ is weak and thus norm dense in X∗ . Next, since ball X is metrizable in the weak∗ -topology for any x ∈ ball X there is a sequence xn0 ∈ ball X0 with xn0 → x, so t → Tt (x) is weak∗ -measurable and t → Tt,∗ (ϕ) is weakly measurable. By the Pettis theorem we conclude that this map is strongly measurable. Finally we apply the Hille theorem about the strong continuity of semigroups. 2 Since Theorem 4 only holds under an additional assumption we have to modify the proof of Proposition 8. We strengthen the assumptions E1–E6 as follows: Let S be a complex Hilbert space with norm · ; we consider it as a symplectic space via σ (f, g) = Imf, g as well as a real Hilbert space (SR , ·,·R ) with f, gR = Ref, g. DOI of original article: 10.1016/j.jfa.2008.05.013. * Corresponding author.
E-mail address: [email protected] (M. Hellmich). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.012
2456
P. Blanchard et al. / Journal of Functional Analysis 259 (2010) 2455–2456
E1 The family of maps f → (F μt )(f ), t 0, is equicontinuous with respect to the nuclear topology τn (SR , · ). I.e. for any > 0 there exists δ > 0 and p ∈ Pn (S, · ) such that |1 − (F μt )(f )| < for any t 0 provided p(f ) < δ. E2 t → (F μt )(f ) is continuous for any f ∈ S. E3 The map S f → Wω (f ) is continuous when S is endowed with the norm topology and M with the strong operator topology. E4–E6 remain unchanged. ∗ group of automorProof of Proposition 8. Since {βt }t0 extends to a weak -continuous phisms on M it remains to consider Vt (x) = S Wω (g)xW (g)∗ dνt (g). By E3 the map g → Wω (g)xWω (g)∗ is weak∗ -continuous for all x ∈ M. The quasimeasure νt is σ -additive by E1 and thus extends to a measure. Now we have f, gR = −σ (if, g), thus eif,gR dνt (g) − 1 = eiσ (−if,g) dνt (g) − 1 = (F μt )(−if ) − 1 < SR
S
provided p(f ) = p(−if ) < δ. Hence the Fourier transforms νˆ t are equicontinuous at 0 with respect to the τn (SR , · )-topology on (SR , ·,·R ). Now by Theorem VI.2.3. of [39] the set {νt }t0 is relatively weakly compact in M1+ (S). By E2 the map t → νˆ t (f ) is continuous for any f ∈ SR , so we obtain that t → νt is weakly continuous by a standard result, i.e. t → ψ dνt is continuous for any bounded · -continuous function ψ . It follows that Vt (x) → x relative to the weak∗ -topology as t ↓ 0 for any x ∈ M. We conclude that t>0 ker Vt = {0}, so Tt = βt ◦ Vt is a weak∗ -continuous semigroup on M by Theorem 4. 2 We remark that it can be shown that with the original assumptions E1–E6 in the paper the conclusion of Proposition 8 holds if we additionally assume that limt↓0 S cω (g + g ) dνt (g) = cω (g ) for any g ∈ S, where cω is the generating functional of ω. With these changes all further results in the paper remain true. References [39] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.