A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X X T T T * T Landau [7] Gelfand 3 * X Banach T sectorial operator ( ) [0] T sectorial operator [2] T S (t) Hilbert e σ(t)t = σ(s (t)) e σ(t)t σ(s (t)) S (t) σ(t) T D(T) X S (t) X D(T) T D(T) [2]
Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ, ψ lim e(λ) +0 λ ix ϕ(x)ψ(x)dx. *2 λ ϕ, ψ ((λ M) ϕ, ψ) ϕ(x)ψ(x)dx + 2πϕ(iλ)ψ(iλ) λ ix ϕ ψ (λ; ϕ, ψ) = ψ(x)ϕ(x)dx + 2πψ(iλ)ϕ(iλ) (e(λ) < 0), λ ix lim e(λ) +0 λ ix ψ(x)ϕ(x)dx (λ i), ψ(x)ϕ(x)dx (e(λ) > 0), λ ix *2 lim e(λ) +0 e(λ) e(λ) 2 ϕ(x)ψ(x)dx = πϕ(im(λ))ψ(im(λ)) + (Im(λ) x) 2 2
ϕ ψ λ X L 2 () X X X ϕ X (λ; ϕ, ψ) ϕ (λ; ϕ, ψ) X (λ;, ψ) X ψ (λ;, ψ) X X A(λ) A(λ)ψ ϕ = ψ(x)ϕ(x)dx + 2πψ(iλ)ϕ(iλ) (e(λ) < 0), λ ix lim e(λ) +0 λ ix ψ(x)ϕ(x)dx (λ i), ψ(x)ϕ(x)dx (e(λ) > 0). λ ix (.2) X X Dirac λ A(λ) = (λ M) A(λ) M M (λ M) L 2 () L 2 () X X λ (X - ) X L 2 () L 2 () X L 2 () Hilbert X L 2 () X (.3) 3 Gelfand 3 rigged Hilbert space H Hilbert T T (λ T) T X X X (λ T) λ λ iemann λ iemann 3
X Banach T C 0 e Tt Laplace e Tt = lim y 2πi x+iy x iy e λt (λ T) dλ (.4) x T ( (a)) x (a) (b) Fig. T ( ) (b) M : ϕ(x) ixϕ(x) e λt (λ T) M X X A(λ) Laplace e Tt = lim y 2πi x+iy x iy e λt A(λ)dλ (.5) ( A(λ) 2 iemann ) (Tϕ)(x) = ixϕ(x) + K ϕ(x)dx (.6) 4
L 2 (, g(x)dx) T (.) K > 0 g ( ) [3, 4] 4 T K c = 2/π K T M K > K c λ = λ(k) K K = K c 0 < K < K c T T ( ϕ, ψ ((λ T) ϕ, ψ) ) 2 T 2 iemann Laplace 2 iemann (.) u(t) X [3, 4] Fig. 2 K K > K c 0 < K < K c iemann D( ) ( ) 2 Gelfand 3 X C Hausdorff X X X µ X ϕ X µ(ϕ) Dirac µ ϕ. a, b C, ϕ, ψ X µ, ξ X 5
µ aϕ + bψ = a µ ϕ + b µ ψ, (2.) aµ + bξ ϕ = a µ ϕ + b ξ ϕ, (2.2) X ( * ) ( * ) ϕ X µ j ϕ µ ϕ {µ j } X µ X X µ j ϕ µ ϕ {µ j } X µ X H Hilbert (, ) X H H H X Hilbert H X 2.. Hausdorff X Hilbert H X H 3 X H X (2.3) rigged Hilbert space Gelfand 3 i : H X ; ψ H i(ψ) ψ i(ψ)(ϕ) = ψ ϕ = (ψ, ϕ), ϕ X (2.4) i : H X X H i ( ) X H (Tréves [8]). 2. i Gelfand 3 Schwartz Gelfand [8] X = C 0 (m ), H = L 2 ( m ) Gelfand 3 Schwartz 3 Gelfand 3 Chiba [5] 3. H C Hilbert H H {E(B)} B B H H = ωde(ω) 6
I Fig. 3 E[ψ, ϕ](ω) Ω. K H T := H + K Schrödinger H K Ω C Ĩ Ĩ I ( 3) T = H + K C X(Ω) (X) X(Ω) H. (X2) X(Ω) H. (X3) X(Ω). (X), (X2) Gelfand 3 X(Ω) H X(Ω) (3.) ( Tréves [8] ) Fréchet Banach Hilbert Fréchet C Montel *3 Banach-Steinhaus *4 *3 Montel Montel C C C Montel Schwartz Montel Montel [9, 3] *4 Banach-Steinhaus. X X X A 4 (i) A (ii) A (iii) A 7
[5] H E(B) (X4) ϕ X(Ω) (E(B)ϕ, ϕ) I E[ϕ, ϕ](ω) Ω I. (X5) λ I Ω E[, ](λ) : X(Ω) X(Ω) C (X4) ϕ, ψ X(Ω) (E(B)ϕ, ψ) I E[ϕ, ψ](ω) d(e(ω)ϕ, ψ) = E[ϕ, ψ](ω)dω, ω I. (3.2) E[ϕ, ψ](ω) ω I Ω I ω ix(ω) X(Ω) X(Ω) A(λ) : ix(ω) X(Ω) λ ω E[ψ, ϕ](ω)dω + 2π E[ψ, ϕ](λ) (λ Ω), A(λ)ψ ϕ = lim y 0 x + E[ψ, ϕ](ω)dω (λ = x I), (3.3) y ω E[ψ, ϕ](ω)dω (Im(λ) < 0), λ ω λ A(λ)ψ ϕ {Im(λ) < 0} Ω I Im(λ) < 0 A(λ)ψ ϕ = ((λ H) ψ, ϕ) A(λ) H A(λ) X(Ω) (λ H) Ω *5 H A(λ) ix(ω) X(Ω) X(Ω) A(λ) i : X(Ω) X(Ω) *6 Q X(Ω) Q : D(Q ) X(Ω) Q D(Q ) X(Ω) C ϕ µ Qϕ µ X(Ω) (iv) A Tréves [8] X Banach *5 Ω A(λ) Ω iemann 4 *6 X(Ω) Banach Banach [] 8
Q µ ϕ = µ Qϕ Q H Hilbert Q Q (Qϕ, ψ) = (ϕ, Q ψ) Q X(Ω) (Q ) Q Q = (Q ) i Q = Q i D(Q) Q Q H K (X6) H X(Ω) X(Ω) Y HY X(Ω) (X7) K H- K X(Ω) (X8) λ {Im(λ) < 0} I Ω K A(λ)iX(Ω) ix(ω) (X6) (X7) H, K, T X(Ω) D(H ) id(h) K, T H, K X(Ω) H, K T X(Ω) K H- K(λ H) H A(λ) (λ H) (X8) (X7) T T (λ T)v = 0 T = H + K (id (λ H) K)v = 0 X(Ω) (λ H) A(λ) 3.. λ Ω I {λ Im(λ) < 0} (id A(λ)K )µ = 0 (3.4) µ X(Ω) λ T µ K (id K A(λ))K µ = 0 (3.5) K µ = 0 (3.4) µ = 0 λ id K A(λ) ix(ω) (X8) K A(λ) ix(ω) well-defined 3.2. λ T µ T µ = λµ (3.6) 9
. D(λ H ) (A(λ)) (λ H )A(λ) = id : ix(ω) ix(ω) (λ H )(id A(λ)K )µ = (λ H K )µ = (λ T )µ = 0 λ T T X(Ω) C T T T 3.2 A(λ) A(λ) n =, 2, A (n) (λ) : ix(ω) X(Ω) A (n) (λ)ψ ϕ = (λ ω) n E[ψ, ϕ](ω)dω + 2π ( )n d n z=λ E[ψ, ϕ](z), (λ Ω), (n )! dz n lim y 0 (x + E[ψ, ϕ](ω)dω, (λ = x I), y ω) n (λ ω) n E[ψ, ϕ](ω)dω, (Im(λ) < 0) (3.7) A (n) (λ)ψ ϕ ((λ H) n ψ, ϕ) Ω A () (λ) A(λ) 3.3. j n 0 A ( j) (λ) (i) (λ H ) n A ( j) (λ) = A ( j n) (λ) A (0) (λ) := id. (ii) A ( j) (λ)(λ H ) n = A ( j n) (λ). (λ H )µ ix(ω) A(λ)(λ H )µ = µ (iii) d j dλ j A(λ)ψ ϕ = ( ) j j! A ( j+) (λ)ψ ϕ, j = 0,,. (iv) ψ X(Ω) A(λ)ψ A(λ)ψ = (λ 0 λ) j A ( j+) (λ 0 )ψ, (3.8) j=0 X(Ω) 0
. (i),(ii) (iii) A(λ) A(λ)ψ ϕ (iii) A(λ)ψ ϕ = (λ 0 λ) j A ( j+) (λ 0 )ψ ϕ, (3.9) j=0 A(λ)ψ X(Ω) Banach-Steinhaus (iv) λ (λ T) n v = 0 n = 2 (λ H K)(λ H K)v = (λ H) 2 (id (λ H) 2 K(λ H)) (id (λ H) K)v = 0. (λ H) 2 (id (λ H) 2 K(λ H)) (id (λ H) K)v = 0. (λ H) n A (n) (λ) (id A (2) (λ)k (λ H )) (id A(λ)K ) µ = 0. B (n) (λ) : D(B (n) (λ)) X(Ω) X(Ω) B (n) (λ) = id A (n) (λ)k (λ H ) n (3.0) B (2) (λ)b () (λ)µ = 0 B (n) (λ) A (n) (λ)k (λ H ) n (λ H ) k B ( j) (λ) = B ( j k) (λ)(λ H ) k, j > k (3.) 3.4. T λ V λ = Ker B (m) (λ) B (m ) (λ) B () (λ). (3.2) m dimv λ λ Ker B () (λ) 3. 3.2 3.5. µ V λ M (λ T ) M µ = 0. V λ m Ker (λ T ) m X(Ω) m Ker (λ T ) m V λ
3.3 λ = (λ T) T λ ψ = (λ H) ( id K(λ H) ) ψ (3.3) X(Ω) (λ H) A(λ) ˆΩ = Ω I {λ Im(λ) < 0}. 3.6. (id K A(λ)) T λ : ix(ω) X(Ω) λ = A(λ) (id K A(λ)) = (id A(λ)K ) A(λ), λ ˆΩ (3.4) 2 (id A(λ)K )A(λ) = A(λ)(id K A(λ)) id K A(λ) ix(ω) id A(λ)K (A(λ)) A(λ) λ A(λ) i λ i : X(Ω) X(Ω) 3.7. 2 λ ˆΩ ˆϱ(T) λ V λ ˆΩ (i) λ V λ λ i X(Ω) X(Ω) X(Ω) (ii) ψ X(Ω) { λ i(ψ)} λ V λ X(Ω) *7 ˆσ(T) := ˆΩ\ˆϱ(T) T ˆσ p (T) id K A(λ) λ ˆσ(T) ( ) ˆσ r (T) λ i X(Ω) λ ˆσ(T) ˆσ c (T) = ˆσ(T)\( ˆσ p (T) ˆσ r (T)) ˆϱ(T) ˆϱ(T) X(Ω) Banach Banach [9, 4] X(Ω) Banach i K A(λ)i X(Ω) λ ˆϱ(T) id i K A(λ)i X(Ω) (.3.5) ˆϱ(T) *7 Banach-Steinhaus 2
3.8. (i) ψ X(Ω) λ iψ ˆϱ(T) X(Ω) - (ii) Im(λ) < 0 λ i = i (λ T) (ii) Im(λ) < 0 ψ, ϕ X(Ω) λ ψ ϕ = ((λ T) ψ, ϕ) λ ψ ϕ ((λ T) ψ, ϕ). ψ λ = i (id K A(λ)) i(ψ) λ+h i(ψ) λ i(ψ) = (A(λ + h) A(λ))i(ψ λ ) + λ+h i i K (A(λ + h) A(λ))i(ψ λ ) h 0 X(Ω) 0 A(λ) i λ 2 λ+h i i K A(λ)i (ii) Banach-Steinhaus { λ i} λ V λ λ+h i h 0 A(λ) i K A(λ)i X(Ω) λ+h i(ψ) λ i(ψ) h λ i(ψ) X(Ω) 3.9. (i) (λ T ) λ = id ix(ω) (ii) µ X(Ω) (λ T )µ ix(ω) λ (λ T )µ = µ. (iii) T λ = λ T. 3.3 (iii) well-defined 3.4 Σ ˆσ(T) γ Ω I {λ Im(λ) < 0} Π Σ : ix(ω) X(Ω) Π Σ ϕ = 2π λ ϕ dλ, ϕ ix(ω), (3.5) γ 3
Pettis *8 Π Σ Π Σ Π Σ Π Σ Σ 3.0. Π Σ (ix(ω)) (id Π Σ )(ix(ω)) = {0} ix(ω) Π Σ (ix(ω)) (id Π Σ )(ix(ω)) X(Ω) (3.7) ϕ X(Ω) µ, µ 2 X(Ω) ϕ i(ϕ) = ϕ = µ + µ 2, µ Π Σ (ix(ω)), µ 2 (id Π Σ )(ix(ω)) (3.8) 3.. Π Σ T - : Π Σ T = T Π Σ. 3.2. λ 0 Π 0 λ 0 V 0 = m Ker B (m) (λ 0 ) B () (λ 0 ) λ 0 Π 0 ix(ω) Π 0 ix(ω) = V 0. Π Π = Π Π λ ( ) 3.2 λ 0 λ = j= (λ 0 λ) j E j E = Π 0 id = (λ T ) λ {E j } j E 3.5 3.2 ˆσ p (T) σ p (T ) ˆσ(T) σ(t) 3.3. C = {Im(λ) < 0} σ p (T) σ(t) T H (i) ˆσ(T) C σ(t) C ˆσ p (T) C σ p (T) C *8 X X S Hausdorff µ S Borel f : S X ϕ X I( f ) ϕ = f ϕ dµ (3.6) S I( f ) X f Pettis I( f ) = f dµ f Pettis S X f Pettis [5] 4
(ii) Σ C σ(t) γ σ(t) γ ˆσ(T) λ C σ(t) λ ˆσ(T). λ C λ i = i (λ T) ( 3.8) λ (λ T) (i) X(Ω) ˆσ(T) ˆσ(T; X(Ω)) 2 X (Ω) X 2 (Ω) (X) (X8) 2 ˆσ(T; X (Ω)), ˆσ(T; X 2 (Ω)) 3.4. X 2 (Ω) X (Ω) X 2 (Ω) X (Ω) (i) ˆσ(T; X 2 (Ω)) ˆσ(T; X (Ω)). (ii) Σ ˆσ(T; X (Ω)) γ ˆσ(T; X (Ω)) γ ˆσ(T; X 2 (Ω)) λ ˆσ(T; X (Ω)) λ ˆσ(T; X 2 (Ω)).. λ X (Ω) X (Ω) X 2 (Ω) X 2 (Ω) (i) (ii) Π Σ Π Σ ix (Ω) {0} X 2 (Ω) X (Ω) Π Σ ix 2 (Ω) {0} X(Ω) *9 X X 2 L * 0 U X LU X 2 L = L(λ) λ L(λ) λ U λ X Banach L(λ) λ L(λ) λ (U ) L U X LU X 2 *9 Schrödinger (resonance pole) [5] complex deformation [] Gelfand 3 *0 Banach 5
L = L(λ) λ L(λ) λ U λ X Banach L(λ) λ ( )L(λ) λ X 2 Montel Montel ( ) ( ) i K A(λ)i 3.5. λ ˆΩ U λ ˆΩ i K A(λ )i : X(Ω) X(Ω) λ U λ id i K A(λ)i X(Ω) λ ˆσ(T). λ i = A(λ) i (id i K A(λ)i) A(λ) i λ V λ {(id i K A(λ )i) ψ} λ V λ X(Ω) λ (id i K A(λ )i) ψ λ V λ id i K A(λ)i λ λ X(Ω) Banach Neumann Banach Neumann Bruyn [2] X(Ω) Banach i K A(λ)i X(Ω) λ ˆϱ(T) id i K A(λ)i X(Ω) i K A(λ)i 3.6. i K A(λ)i : X(Ω) X(Ω) λ ˆΩ (i) D ˆΩ D ˆσ p (T) ˆΩ (ii) 3.2 (iii) ˆσ c (T) = ˆσ r (T) =. X(Ω) Banach iesz-schauder X(Ω) Banach iesz-schauder (ingrose [6]) T H T 6
3.7. T 3.6 I T (H ) I. λ 0 I T (H ) T λ 0 P 0 ϕ = lim ε 0 ε (λ0 + ε T) ϕ, ϕ H, (3.9) H X(Ω) Π 0 i P 0 = Π 0 i P 0 H Π 0 ix(ω) σ p (T) ˆσ p (T) 3.6 ˆσ p (T) 3.6 T = (H +K) H C 0 - e Tt Laplace (e Tt ψ, ϕ) = 2π lim x x y x y e λt ((λ T) ψ, ϕ)dλ, x, y, (3.20) T T t 3.8 ϕ, ψ X(Ω) (e Tt ψ, ϕ) = 2π lim x x y x y e λt λ ψ ϕ dλ, (3.2) 3.6 λ ψ ϕ Π 0 λ 0 M 2π γ 0 e λt λ ψ ϕ dλ = M k=0 e λ0 t ( t) k (λ 0 T ) k Π 0 ψ ϕ, k! λ 0 0 T ( T 7
) (e Tt ψ, ϕ) e Tt ψ H Landau [7] Schrödinger [, 5] λ 0 µ 0 X(Ω) (e Tt ) = ((e Tt ) ) (e Tt ) µ 0 = e λ 0 t µ 0 µ 0 µ 0 X(Ω) X(Ω) T > 0 ε > 0 ϕ 0 X(Ω) 0 t T (e Tt ) ϕ 0 ψ (e Tt ) µ 0 ψ < ε, 0 t T (e Tt ϕ 0, ψ) e λ0 t µ 0 ψ, (3.22) t 4 4. [3, 4] g (z) < ω < g (ω) > 0 g(ω) g(ω) = 0 (ω < ), g (ω) ( < ω < ), 0 (ω > ), (4.) H = L 2 (, g(ω)dω) L 2 H (Hϕ)(ω) = ωϕ(ω) H H σ(h) supp(g) = [, ] (E(ω)ψ, ϕ) := E[ψ, ϕ](ω) = 0 (ω < ), ψ(ω)ϕ(ω)g (ω) ( < ω < ), 0 (ω > ), (4.2) X L 2 (, g(ω)dω) 8
g(ω) ψ, ϕ X ((λ H) ψ, ϕ) ω < ω > E[ψ, ϕ](ω) 0 ((λ H) ψ, ϕ) ((λ H) ψ, ϕ) < ω < ((λ H) ψ, ϕ) A(λ)ψ ϕ = λ ω ψ(ω)ϕ(ω)g (ω)dω + 2π ψ(λ)ϕ(λ)g (λ), (4.3) < ω < + n ((λ H) ψ, ϕ) λ ω ψ(ω)ϕ(ω)g (ω)dω + 2π n ψ(λ)ϕ(λ)g (λ), (λ H) X X A(λ) iemann ± P 0 (ω) L 2 (, g(ω)dω) K (Kϕ)(ω) = κ(ϕ, P 0 )P 0 (ω) κ > 0 T := H + K g (.6) K T H σ c (T) = [, ] λ ω g(ω)dω = 0, (4.4) κ λ = x + y, x, y x ω y g(ω)dω = 0, (x ω) 2 + y2 (x ω) 2 + y g(ω)dω = 2 κ. (4.5) g(ω) x = 0 x = 0 κ > 0 κ κ κ y 0 y lim y 0 ω 2 + y g(ω)dω = πg(0) = 2 κ. (4.6) κ (πg(0)) λ(κ) 0 κ = (πg(0)) σ c (T) = [, ] 9
T X 3.6 ˆσ c (T) = ˆσ r (T) = 2 iemann 2 iemann λ ω g (ω)dω + 2π g (ω) = 0 (4.7) κ (4.4) κ (πg(0)) (4.4) (4.7) u/ t = Tu e Tt 3.6 Laplace 4 Fig. 4 γ Laplace γ [, ] iemann iemann 2 ± γ 3.6 g(ω) g(ω) ( (4.2) 2 ω ) γ u(t) X t 0 [3, 4] 20
4.2 Schrödinger [6] Schrödinger T = + V m V V : m C ( ) H = L 2 ( m ), V H, K H (λ H) ψ(x) = (2π) m/2 m λ ξ e x ξ F [ψ](ξ)dξ, 2 F Fourier S m m m ξ m ξ = rω, r 0, ω S m (λ H) ψ(x) (λ H) ( r m 2 ψ(x) = e rx ω F [ψ]( ) rω)dω dr, (4.8) (2π) m/2 λ r 2 0 S m arg(λ) = 0 H {λ 2π < arg(λ) < 0} L 2 ( m )- f (z) := F [ψ]( zω) λ ( r m 2 e rx ω F [ψ]( ) rω)dω dr (2π) m/2 0 λ r S m 2 + π λ m 2 λx ω F [ψ]( λω)dω, (4.9) (2π) m/2 S m e V X(Ω) a > 0 V e 2a x V(x) L 2 ( m ) (4.0) a > 0 X(Ω) := L 2 ( m, e 2a x dx) X(Ω) L 2 ( m, e 2a x dx) Gelfand 3 L 2 ( m, e 2a x dx) L 2 ( m ) L 2 ( m, e 2a x dx) (4.) T = + K (X) (X8) 3.6 ψ L 2 ( m, e 2a x dx) r F [ψ](rω) {r C a < Im(r) < a} λ F [ψ]( λω) iemann P(a) = {λ a < Im( λ) < a} ψ L 2 ( m, e 2a x dx) (4.9) iemann ( z = 0 ) m [6] [6] 2
4.3 Evans [6] Evans E(λ) Evans P Fredholm E(λ) P [7] Evans P P (X) (X8) Gelfand 3 E(λ) [6] [] J. Bonet, On the identity L(E, F) = LB(E, F) for pairs of locally convex spaces E and F, Proc. Amer. Math. Soc. 99 (987), no. 2, 249-255 [2] G. F. C. de Bruyn, The existence of continuous inverse operators under certain conditions, J. London Math. Soc. 44 (969), 68-70 [3] H.Chiba, I.Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators, Chaos, 2, 04303 (20) [4] H. Chiba, A proof of the Kuramoto s conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, (submitted, arxiv:008.0249) [5] H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under certain analyticity conditions, (submitted, arxiv:07.5858) [6] H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under certain analyticity conditions: applications to Schrödinger operators, (submitted) [7] J. D. Crawford, P. D. Hislop, Application of the method of spectral deformation to the Vlasov-Poisson system, Ann. Physics 89 (989), no. 2, 265 37 [8] I. M. Gelfand, N. Ya. Vilenkin, Generalized functions. Vol. 4. Applications of harmonic analysis, Academic Press, New York-London, 964 [9] A. Grothendieck, Topological vector spaces, Gordon and Breach Science Publishers, New York-London-Paris, 973 [0] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, (98) [] P. D. Hislop, I. M. Sigal, Introduction to spectral theory. With applications to Schrodinger operators, Springer-Verlag, New York, 996 [2] W. Kerscher,. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Appl. Math. 2 (984), 297-309. [3] H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 9, (967), 366 383 [4] F. Maeda, emarks on spectra of operators on a locally convex space, Proc. Nat. Acad. Sci. U.S.A. 47, (96) [5] M. eed, B. Simon, Methods of modern mathematical physics IV. Analysis of operators, Academic Press, New York-London, 978 [6] J.. ingrose, Precompact linear operators in locally convex spaces, Proc. Cambridge Philos. Soc. 53 (957), 58-59 [7] B. Sandstede, Stability of travelling waves, Handbook of dynamical systems, Vol. 2, 983-055, North-Holland, Amsterdam, 2002 [8] F. Tréves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 967 [9] L. Waelbroeck, Locally convex algebras: spectral theory, Seminar on Complex Analysis, Institute of Advanced Study, 958 22