Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus

We give a direct non-abstract proof of the spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the spectral mapping theorem for self-adjoint operators on Hilber...

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1. Verfasser:	Claire, N.S.
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spelling	irk-123456789-1067202016-10-04T03:02:26Z Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus Claire, N.S. We give a direct non-abstract proof of the spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the spectral mapping theorem for self-adjoint operators on Hilbert spaces. Our exposition is closer in spirit to the proof by explicit construction of the existence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to derive a functional calculus for semi-bounded operators. Представлено прямое неабстрактное доказательство теоремы об отображении спектра для функционального исчисления Девиса-Хельффера-Сьостранда для линейных операторов в банаховых пространствах с реальными спектрами, а следовательно, дано новое неабстрактное прямое доказательство теоремы об отображении спектра для самосопряженных операторов в гильбертовых пространствах. Наше представление по духу ближе к доказательству благодаря явной конструкции существования функционального исчисления, данной Дэвисом. Мы применяем теорему расширения Сили, чтобы получить функциональное исчисление для полуограниченных операторов. 2012 Article Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus / N.S. Claire // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 221-239. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106720 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	We give a direct non-abstract proof of the spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the spectral mapping theorem for self-adjoint operators on Hilbert spaces. Our exposition is closer in spirit to the proof by explicit construction of the existence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to derive a functional calculus for semi-bounded operators.
format	Article
author	Claire, N.S.
spellingShingle	Claire, N.S. Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus Журнал математической физики, анализа, геометрии
author_facet	Claire, N.S.
author_sort	Claire, N.S.
title	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus
title_short	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus
title_full	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus
title_fullStr	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus
title_full_unstemmed	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus
title_sort	spectral mapping theorem for the davies-helffer-sjöstrand functional calculus
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/106720
citation_txt	Spectral Mapping Theorem for the Davies-Helffer-Sjöstrand Functional Calculus / N.S. Claire // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 221-239. — Бібліогр.: 8 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT clairens spectralmappingtheoremforthedavieshelffersjostrandfunctionalcalculus
first_indexed	2025-07-07T18:54:00Z
last_indexed	2025-07-07T18:54:00Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2012, vol. 8, No. 3, pp. 221–239 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus Narinder S. Claire Global Equities & Commodity Derivatives Quantitative Research BNP Paribas London 10 Harewood Avenue, London NW1 6AA E-mail: narinder.claire@uk.bnpparibas.com Received December 21, 2010, revised February 20, 2012 We give a direct non-abstract proof of the spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the spectral mapping theorem for self-adjoint operators on Hilbert spaces. Our exposition is closer in spirit to the proof by explicit construction of the existence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to derive a functional calculus for semi-bounded operators. Key words: Functional calculus, spectral mapping theorem, spectrum. Mathematics Subject Classification 2010: 47A60. 1. Introduction The Helffer–Sjöstrand formula was established in [1] in the following propo- sition: Proposition 1.1. ([1] Proposition 7.2) Let H be a self-adjoint operator (not necessarily bounded) on a Hilbert space H. Suppose f is in C∞ 0 (R) and f̃ in C∞ 0 (C) is an extension of f such that ∂f̃ ∂z̄ = 0 on R. Then we have f (H) = − 1 π ∫∫ C ∂f̃ (z) ∂z (z −H)−1 dxdy, (1.1) where z = x+ iy. c© Narinder S. Claire, 2012 Narinder S. Claire The existence of the functional calculus was assumed by the authors. Davies [2] showed that the formula (Equation 1.1) yielded a new approach to constructing the functional calculus for linear operators on Banach spaces under the following hypothesis: Hypothesis 1.2. H is a closed densely defined operator on a Banach space B with spectrum σ(H) ⊆ R. The resolvent operators (z −H)−1 are defined and bounded for all z /∈ R and ‖(z −H)−1‖ ≤ c\|Im z\|−1 ( 〈z〉 \|Im z\| ) α (1.2) for some α ≥ 0 and all z /∈ R, where 〈z〉 := (1 + \|z\|2) 1 2 . His functional calculus for operators on Banach spaces was defined for an algebra of slowly decreasing smooth functions. Davies [2] pointed out that a functional calculus based upon almost analytic extensions was also constructed by Dyn’kin [3]. However, the two approaches were quite different and that Davies’ approach was more appropriate for differential operators. A spectral mapping theorem for the Davies–Helffer–Sjöstrand functional cal- culus was proved by Bátkai and Fašanga [4]. They applied methods from abstract functional analysis and their primary tool was an existing abstract spectral map- ping theorem from the theory of Banach algebras: Theorem 1.3. ([4] Theorem 4.1) Let B1 be a commutative, semisimple, reg- ular Banach algebra, B2 be a Banach algebra with a unit, Θ : B1 → B2 be a continuous algebra homomorphism and a ∈ B1. Then σB2 (Θ (a)) = â (Sp (θ)) , where Sp (Θ) := ∩b∈KerΘKer b̂ andˆdenotes the Gelfand transform. Our exposition of the spectral mapping theorem, part of the Ph.D thesis re- ferred to in the introduction of [4], takes a very non-abstract and direct approach to the proof. In particular, an existing spectral mapping is not assumed. Our sole ingredients, supplementing the tools provided by Davies in [2], are the very elementary observations: • ([5] Problem 8.1.11) If H is a closed operator and λ lies in the topological boundary of the spectrum of H, then for every ǫ > 0 there is a vector v with norm 1 such that ‖Hv − λv‖ < ǫ. • Stokes’ Formula has similarities to the Cauchy Integral Formula. 222 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus In the last part of our exposition we derive a functional calculus for operators with spectra bounded on one side. Our main tool here is an extension operator of Seeley, E : C∞[0,∞) −→ C∞ (R) . 1.1. Functional Calculus We summarize some of the main aspects of the Davies–Helffer–Sjöstrand func- tional calculus presented in [6] and some properties of the algebra of functions. Let ψa,ǫ be a smooth function such that ψa,ǫ (x) := { 1 if x ≥ a 0 if x ≤ a− ǫ. Then given an interval [a, b] , we define the approximate characteristic function Ψ [a,b],ǫ Ψ [a,b],ǫ (x) = ψa,ǫ (x) − ψ b+ǫ,ǫ (x) which has a support [a− ǫ, b+ ǫ] and is equal to 1 in [a, b] and is smooth. Definition 1.4. For β ∈ R let Sβ be the set of all complex-valued smooth functions defined on R, where for every n ∈ N ∪ {0} there is a positive constant cn such that \| dnf (x) dxn \| ≤ cn〈x〉 β−n. We then define the algebra A := ⋃ β<0 Sβ. Lemma 1.5. (Davies [2, 6]) A is an algebra under point-wise multiplication. For any f in A the expression ‖f‖n := n∑ r=0 ∞∫ −∞ \| drf (x) dxr \| 〈x〉r−1dx (1.3) defines a norm on A for each n. Moreover, C∞ 0 (R) is dense in A with this norm. Lemma 1.6. The function 〈x〉β is in A for each β < 0. P r o o f. The statement follows from the observations that if β < 0 and m ≥ n, then xn〈x〉β−m ≤ 〈x〉β and d ( xn〈x〉β−m ) dx = nxn−1〈x〉β−m + (β −m) xn+1〈x〉β−m−2. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 223 Narinder S. Claire Lemma 1.7. Let s ∈ R. If f is in A, then the function gs (x) := { f(x)−f(s) x−s x 6= s f ′ (s) x = s is also in A. P r o o f. When \|x− s\| is large, then 1 \|x− s\| ≤ cs〈x〉 −1 for some cs > 0. Moreover, g(r) s (x) = r∑ m=0 crf (m)(x) (x− s)m−r−1 + cf (s) (x− s)−r−1 and lim x→s g(m) s (x) = 1 m+ 1 f (m+1)(s). Lemma 1.8. If f ∈ Sβ for β < 0 and g ∈ S0, then fg ∈ A. P r o o f. \|(fg)(r)(x)\| ≤ cr r∑ m=0 \|g(r−m)(x)\| \|f (m)(x)\| ≤ cr,φ〈x〉 β−r. The following concept of almost analytic extensions is due to Hörmander [7, p. 63]. Definition 1.9. Let τ (x, y) be a smooth function such that τ (x, y) := { 1 if \|y\| ≤ 〈x〉 0 if \|y\| ≥ 2〈x〉. Then given f ∈ A we define an almost analytic extension f̃ as f̃ (x, y) := ( n∑ r=0 drf (x) dxr (iy)r r! ) τ (x, y) (1.4) for some n ∈ N. Moreover, we define ∂f̃ ∂z := 1 2 ( ∂f̃ ∂x + i ∂f̃ ∂y ) . (1.5) The specific choices of τ and n are not critical. 224 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus The following lemma establishes the construction of the new functional cal- culus: Lemma 1.10. (Davies [2]) Let f ∈ A, then define f (H) := − 1 π ∫∫ C ∂f̃ (z) ∂z (z −H)−1 dxdy, (1.6) where f̃ is an almost-analytic version of f as defined in definition 1.9 and z = x+ iy. Then i. If n > α, then subject to hypothesis 1.2 the integral (1.6) is norm convergent for all f in A and ‖f (H) ‖ ≤ c‖f‖n+1. ii. The operator f (H) is independent of n and the cut-off function τ , subject to n > α. iii. If f is a smooth function of compact support disjoint from the spectrum of H, then f (H) = 0. iv. If f and g are in A, then (fg) (H) = f (H) g (H). v. If z 6∈ R and gz (x) := (z − x)−1 for all x ∈ R, then gz ∈ A and gz (H) = (z −H)−1 . 1.2. Preliminaries Definition 1.11. Given z, ω in C, we define the curve Γ in the complex plane Γ(z, ω, α) := ((1 − α)\|z\| + α\|ω\|) ei(1−α)Arg(z)+iαArg(ω), where α ∈ [0, 1]. The important property of Γ is that it is able to connect two non-zero points in the complex plane without intersection with the origin. Theorem 1.12. Let λ ∈ C. If f is a smooth complex valued function in the interval [a, b], where f (a) 6= λ and f (b) 6= λ, then there is a smooth function h in C∞ ([a, b]) such that {x ∈ [a, b] : h (x) = λ} is empty , and f − h and all derivatives of f − h vanish at a and b. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 225 Narinder S. Claire P r o o f. Let g (x) := Γ ( f (a) − λ, f (b) − λ, x− a b− a ) + λ. (1.7) Since f is continuous, we know there is an 0 < ǫ < b−a 2 such that {x ∈ [a, b]/ (a+ ǫ, b− ǫ) : f (x) = λ} = ∅. Then we can define h := ( 1 − Ψ [a+ǫ,b−ǫ],ǫ ) f + Ψ [a+ǫ,b−ǫ],ǫ g. Lemma 1.13. Given f ∈ A, let λ be a non-zero point in C and let Aλ := {x : f (x) = λ}. If Aλ ∩ σ (H) is empty, then there is a function h ∈ A such that h (x) 6= λ for all x ∈ R and h (H) = f (H) . P r o o f. If Aλ is empty, then we put h = f . If Aλ is not empty, then Aλ is a compact subset of ρ (H). Moreover, Aλ can be covered by a finite set of closed disjoint intervals [ai, bi] which are also subsets of ρ (H). By applying Theorem 1.12 to each interval, we can find a function h in A such that h(x) = f(x) for all x ∈ σ (H) and h(x) 6= λ for all x ∈ R. Moreover, since (f − h) has a compact support in ρ (H) , then it follows from Lemma 1.10 (iii) that h (H) = f (H). 2. Bounded Operators We let B be a bounded operator satisfying hypothesis (1.2). Moreover, let u := supσ (B) and l := inf σ (B) . Lemma 2.1. For any f ∈ A and ǫ > 0 fΨ [l′,u′],ǫ (B) = f (B) , where l′ < l and u′ > u. 226 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus P r o o f. Suppose f has a compact support, then f − fΨ [l′,u′],ǫ has a compact support disjoint from the spectrum of B, hence, by Lemma 1.10 (iii), the statement of the lemma is true for functions in C∞ 0 (R). The statement for all f ∈ A follows from the density of C∞ 0 (R) in A. Lemma 2.2. Let f ∈ A. If ǫ > 0 and Dǫ := {z : \|z − u+l 2 \| < u−l 2 + ǫ} and ∂Dǫ := {z : \|z − u+l 2 \| = u−l 2 + ǫ}, then f (B) = 1 2πi ∫ ∂Dǫ f̃ (z) (z −B)−1 dz − 1 π ∫ Dǫ ∂f̃ ∂z (z −B)−1 dxdy. P r o o f. By Lemma 2.1, we can assume that f has a compact support in [l − ǫ, u+ ǫ] . If R > u−l 2 + ǫ and AR is the annulus {z : u−l 2 + ǫ < \|z − u+l 2 \| < R}, then ∫ \|z− u+l 2 \|<R ∂f̃ ∂z (z −B)−1 dxdy = ∫ AR ∂f̃ ∂z (z −B)−1 dxdy + ∫ Dǫ ∂f̃ ∂z (z −B)−1 dxdy. Applying Stokes’ theorem ∫ AR ∂f̃ ∂z (z −B)−1 dxdy = 1 2i ∫ \|z− u+l 2 \|=R f̃ (z −B)−1 dz − 1 2i ∫ ∂Dǫ f̃ (z −B)−1 dz and letting R be large enough for f̃ to vanish on {z : \|z − u+l 2 \| = R} completes the proof. Lemma 2.3. Let ǫ > 0. If l′ < l and u′ > u, then Ψ [l′,u′],ǫ (B) = 1. P r o o f. Let 0 < δ < 1, and define Ω as the open rectangle {z ∈ C : \|Re z − u′+l′ 2 \| < u′−l′ 2 , \|Im z\| < δ}, as illustrated in Fig. 1. Using a similar argument to that given in the proof of Lemma 2.2, we see that Ψ [l′,u′],ǫ (B) = 1 2πi ∫ ∂Ω Ψ̃ [l′,u′],ǫ (z, z) (z −B)−1 dz − 1 π ∫ Ω ∂Ψ̃ [l′,u′],ǫ ∂z (z −B)−1 dxdy. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 227 Narinder S. Claire When l′ ≤ x ≤ u′, then Ψ [l′,u′],ǫ (x) = 1. Moreover, when l′ ≤ x ≤ u′, then Ψ(n) [l′,u′],ǫ (x) = 0 for all n > 0. Recalling definition (1.4), we can see that Ψ̃ [l′,u′],ǫ (z, z) = 1 for all z ∈ Ω, hence Ψ [l′,u′],ǫ (B) = 1 2πi ∫ ∂Ω (z −B)−1 dz and we conclude with an application of Cauchy’s integral formula. u+l 2 −R l 0 u u+l 2 +R −iR −iδ i δ iR � x� −�x� � Fig. 1. Integral domain for Lemma 2.3. 228 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus 3. Enlargement of A We extend the algebra A of slow decaying functions in a trivial but necessary way. The purpose of the extension is to provide a multiplicative identity, the constant 1 function. Definition 3.1. Let Â := {(z, f) : z ∈ C, f ∈ A}, where for each x ∈ R we define (z, f) (x) := z + f(x). Moreover, we define the point-wise addition and multiplication: (ω, f) ◦ (z, g) := (ωz, ωg + zf + fg) , (ω, f) + (z, g) := (ω + z, f + g) . It is clear that (1, 0) , the multiplicative identity, and (0, 0) , the additive identity, are in Â, and the algebra is closed under these operations. For any z ∈ C we will denote (z, 0) ∈ Â simply by z. Given φ = (z, f) ∈ Â, let π A,φ := f and π C,φ := z and let ‖φ‖n := \|π C,φ \| + ‖π A,φ ‖n. Definition 3.2. We have the extended functional calculus. For φ ∈ Â, let φ(H) := π A,φ (H) + π C,φ I along with the implied norm ‖φ (H) ‖ := \|π C,φ \| + ‖π A,φ (H) ‖ ≤ \|π C,φ \| + k‖π A,φ ‖n+1 ≤ k‖φ‖n+1 for some k > 1. Definition 3.3. For φ ∈ Â, let µ (φ) := 1 π C,φ + π A,φ − 1 π C,φ . Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 229 Narinder S. Claire Lemma 3.4. If φ ∈ Â and −π C,φ is not in Ran(π A,φ ), then µ (φ) is in A, and φ−1 = ( 1 π C,φ , µ (φ) ) . P r o o f. By re-writing µ (φ) = 1 π C,φ + π A,φ − 1 π C,φ = −π A,φ π C,φ ( π C,φ + π A,φ ) , then it is routine exercise in differentiation to show that −1 π C,φ ( π C,φ + π A,φ ) is in S0. Then, since π A,φ is in A, Lemma 1.8 implies the statement. Corollary 3.5. Given φ = (z, f) ∈ Â and λ ∈ C such that − (z − λ) is not in the closure of the range of f, then we have (φ− λ)−1 ∈ Â. 4. Spectral Mapping Theorem Lemma 4.1. If φ is in Â, then σ (φ(H)) ⊆ Ran (φ). P r o o f. Given λ ∈ C which is not in Ran (φ), we have by Corollary 3.5 (φ− λ)−1 ∈ Â, hence (φ (H) − λ)−1 exists and is bounded and therefore λ 6∈ σ (φ (H)). Lemma 4.2. If φ is in Â, then σ (φ (H)) ⊆ φ (σ(H)) ∪ {π C,φ }. P r o o f. Let λ ∈ C be such that λ 6= π C,φ and let Aλ = {x : φ(x) = λ}. If Aλ ∩ σ(H) = ∅, then by Lemma 1.13, we have that there is function h in A such that h(x) = π A,φ (x) for all x ∈ σ (H) , 230 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus and h(x) 6= λ− π C,φ for all x ∈ R, moreover, h (H) = π A,φ (H) . Since θ := ( π C,φ , h ) ∈ Â, it follows from the definition of the enlargement of the algebra that φ (H) = θ (H) . Since λ /∈ Ran (θ), the statement of the lemma follows from Lemma 4.1. Lemma 4.3. Let φ ∈ Â. If H is bounded and {x : φ(x) = π C,φ } ∩ σ(H) is empty , then π C,φ /∈ σ (φ (H)). P r o o f. Let u := supσ (H) and l := inf σ (H). Let 0 < ǫ≪ 1 such that π A,φ is not zero on [l − ǫ, l] and on [u, u+ ǫ]. Then let u′ := u+ ǫ and l′ := l − ǫ. The set {x ∈ [ l′, u′ ] : π A,φ (x) = 0} can be covered by a finite number of disjoint intervals [ai, bi] which are all disjoint from σ (H) and are all in [l′, u′]. Applying Lemma 1.12 to each [ai, bi] , we can find a function f ∈ A such that {x ∈ [ l′, u′ ] : f(x) = 0} = ∅ and f = π A,φ for all x in R/ [l′, u′]. Let g be any function in A such that g (x) = 1 f(x) for all x ∈ [l′, u′] . By Lemma 1.10 (iii), we have π A,φ (H) g (H) = f (H) g (H) and by Lemma 2.1, we have f (H) g (H) = (fgΨ [l′,u′],ǫ )(H) = Ψ [l′,u′],ǫ (H), hence by Lemma 2.3, we have π A,φ (H) g (H) = 1 and, consequently, ( −π C,φ + φ(H) ) g(H) = 1. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 231 Narinder S. Claire Theorem 4.4. If φ in Â, then σ (φ (H)) ⊆ φ (σ(H)). P r o o f. If H is unbounded, then φ (σ(H)) = φ (σ(H)) ∪ {π C,φ } and the theorem follows from Lemma 4.2. If H is bounded and there is an x ∈ σ (H) such that φ (x) = π C,φ , then φ (σ(H)) = φ (σ(H))∪{π C,φ } and again the theorem follows from 4.2. If H is bounded and φ (x) 6= π C,φ for all x ∈ σ (H) , then φ (σ(H)) = φ (σ(H)) by Lemmas 4.2 and 4.3. Lemma 4.5. Given s ∈ R and a function f ∈ A, let ks (x) := ( 1,− s+i x+i ) ∈ Â and let the function gs be defined as in Lemma 1.7, then (f (H) − f (s)) (H + i)−1 = gs (H) ks (H) . P r o o f. This statement follows directly from the functional calculus and the observation (−f (s) , f (x)) ( 0, (x+ i)−1 ) = ( 0, f (x) − f (s) x− s )( 1,− s+ i x+ i ) . Theorem 4.6. Let f be a function in A, then f (σ (H)) ⊆ σ (f (H)) . P r o o f. We observe the identity H − x = (H + i) − (x+ i) = ( 1 − (x+ i) (H + i)−1 ) (H + i) = kx (H) (H + i) (4.1) for some x ∈ R. Let s ∈ R. Suppose there is a sequence of unit norm vectors {vm} ⊂ Dom (H) such that lim m→∞ (H − s) vm = 0. Using identity (4.1 ), we have lim m→∞ gs (H) ks (H) (H + i) vm = 0. By applying Lemma 4.5, we can conclude that lim m→∞ (f (H) − f (s)) vm = 0. The accumulation points of f (σ (H)) are in σ (f (H)) since the latter is closed. R e m a r k 4.7. In the proof of Theorem 4.6, σ (H) is equal to the approximate point spectrum ofH and it is proved that if s is in the approximate point spectrum of H, then f (s) is in the approximate point spectrum of f (H). Corollary 4.8. Let φ be a function Â, then φ (σ (H)) ⊆ σ (φ (H)) . 232 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus 5. Self-Adjoint Operators We now assume that H is self-adjoint and B is a Hilbert space. The following theorem of Davies extends the Davies–Helffer–Sjöstrand functional calculus to C0 (R) for self-adjoint operators. Theorem 5.1. (Davies [2] Theorem 9) The functional calculus may be ex- tended to a map from f ∈ C0 (R) to f (H) ∈ L (B) with the following properties: i. f → f (H) is an algebra homomorphism. ii. f (H) = f (H)∗ . iii. ‖f (H) ‖ ≤ ‖f‖∞. iv. If z 6∈ R and gz (x) := (z − x)−1 for all x ∈ R, then gz (H) = (z −H)−1 . Moreover, the functional calculus is unique subject to these conditions. Lemma 5.2. If f ∈ C0 (R) , then f (σ (H)) ⊆ σ (f (H)) . P r o o f. This is a consequence of the density of A in C0 (R). By the Stone–Weierstrass theorem, the linear subspace { n∑ i=1 λi x−ωi : λi ∈ C ωi /∈ R} is dense in C0 (R). If fǫ ∈ A is close to f and if v ∈ B is of norm 1, then ‖f (H) v − f (s) v‖ ≤ ‖f (H) − fǫ (H) ‖ + ‖fǫ (H) v − fǫ (s) v‖ + ‖fǫ − f‖∞. The statement then follows from Lemma 5.1 (iii). Lemma 5.3. If f ∈ C0 (R) , then σ (f (H)) ⊆ f (σ (H)). P r o o f. Let fn be a sequence converging to f in C0 (R) such that fn (x) := n∑ i=1 λn,i x−ωn,i , ωn,i /∈ R. The existence of such a sequence follows from the Stone–Weierstrass theorem as explained in the proof of Lemma 5.2. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 233 Narinder S. Claire Suppose λ ∈ C is not in the closure of f (σ (H)). Then there is δ > 0 such that inf s∈σ(H) \|f (s) − λ\| = δ. Also for all large enough n, we have ‖fn − f‖∞ < δ 2 . Then from \|f (s) − fn (s) + fn (s) − λ\| ≥ δ we can deduce that \|fn (s) − λ\| > δ − ‖fn − f‖∞, hence inf s∈σ(H) \|fn (s) − λ\| > δ 2 and λ /∈ σ (fn (H)). From the identity ‖ (f (H) − λ) (fn (H) − λ)−1 − 1‖ = ‖ (f (H) − fn (H)) (fn (H) − λ)−1 ‖ we can deduce that λ /∈ σ (f (H)). 6. Functional Calculus for Semi-Bounded Operators We modify our main hypothesis (1.2) by assuming that the spectrum of H is bounded below and, without loss of generality, σ (H) ⊆ [0,∞). We introduce a new ring of functions A+. Definition 6.1. Sβ + is the set of smooth functions on R + ∪{0} with the same decaying property as Sβ, that is, for every n there is positive constant cn such that \| dnf (x) dxn \| ≤ cn〈x〉 β−n. Then A+ is defined appropriately and similarly we define the Banach space A+ n with norm ‖f‖A+ n := n∑ r=0 ∞∫ 0 \| drf (x) dxr \|〈x〉r−1dx. (6.1) We present a theorem due to Seeley [8] which gives a linear extension operator for smooth functions from the half line to the whole line. 234 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus Theorem 6.2. (Seeley’s Extension Theorem) There is a linear extension op- erator E : C∞[0,∞) −→ C∞ (R) such that for all x > 0 (Ef) (x) = f (x) . The extension operator is continuous for many topologies including uniform convergence of each derivative. The proof of the theorem relies on the following lemma. Lemma 6.3. ([8]) There are sequences {ak}, {bk} such that i. bk < −1. ii. ∞∑ k=0 \|ak\|\|bk\| n <∞ for all non-negative integers n. iii. ∞∑ k=0 ak (bk) n = 1 for all non-negative integers n. iv. bk → −∞. The proof to Seeley’s extension theorem is by construction and it is informa- tive to give explicitly the extension. First, we need to define two linear operators. Definition 6.4. Given f ∈ A+, φ ∈ A and real a, we define (Taf) (x) = f (ax) , (Sφf) (x) = φ (x) f (x) . P r o o f. (Proof of Seeley’s Extension Theorem.) Let φ ∈ C∞ c (R) such that φ (x) =    1, x ∈ [0, 1], 0, x ≥ 2, 0, x ≤ −1. Then define E such that (Ef) (x) :=    ∞∑ k=0 ak (Tbk Sφf) (x) , x < 0, f (x) , x ≥ 0. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 235 Narinder S. Claire Lemma 6.5. If a > 1, then ‖Ta‖A+ n →A+ n ≤ an. P r o o f. The proof follows from ‖Taf‖A+ n = n∑ r=0 ∞∫ 0 \| drf (ax) dxr \|〈x〉r−1dx ≤ n∑ r=0 ar ∞∫ 0 \| drf (x) dxr \|〈x〉r−1dx. Lemma 6.6. If φ ∈ A, then Sφ is a bounded operator with respect to each norm ‖ ‖A+ n . P r o o f. A simple application of Leibniz’s rule gives dr (φ (x) f (x)) dxr = r∑ m=0 cr dr−m (φ (x)) dxr−m dm (f (x)) dxm , then \| dr (φ (x) f (x)) dxr \| ≤ cr r∑ m=0 dr−m,φ 〈x〉 β−(r−m) d m (f (x)) dxm ≤ cr,φ r∑ m=0 〈x〉m−r d m (f (x)) dxm we integrate to give ∞∫ 0 \| dr (φ (x) f (x)) dxr \|〈x〉r−1dx ≤ cr,φ r∑ m=0 ∞∫ 0 \| dm (f (x)) dxm \|〈x〉m−1dx = cr,φ‖f‖A+ r , and hence we have our estimate ‖Sφf‖A+ n = n∑ r=0 ∞∫ 0 \| dr (φ (x) f (x)) drx \|〈x〉r−1dx ≤ cn,φ n∑ r=0 ‖f‖A+ r ≤ cn,φ‖f‖A+ n . 236 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus Theorem 6.7. For each normed vector space A+ n Seeley’s Extension Operator is a bounded operator from A+ n to An. P r o o f. ‖Ef‖An = n∑ r=0 ∞∫ −∞ \| dr (Ef) dxr \|〈x〉r−1dx = n∑ r=0 ∞∫ 0 \| drf (x) dxr \|〈x〉r−1dx+ n∑ r=0 0∫ −∞ \| ∞∑ 0 ak dr (φ (bkx) f (bkx)) dxr \|〈x〉r−1dx = ‖f‖A+ n + ‖ ∞∑ k=0 akT−bk Sφf‖A+ n ≤ ‖f‖A+ n + ∞∑ k=0 \|ak\| ‖Sφ‖ ‖\|T−bk ‖‖f‖A+ n ≤ ‖f‖A+ n + ( ∞∑ k=0 \|ak\| \|bk\| n ) cn,φ‖f‖A+ n and hence the extension operator is continuous. If f and g are elements of A such that f \|[0,∞] = g\|[0,∞] and the spectrum of H is [0,∞), then it is not necessary that supp (f − g) ∩ σ (H) is empty since supp (f − g) ∩ σ (H) = {0} is possible. Lemma 6.8. If f is a smooth function on R of a compact support such that supp (f) = [−a, 0] and H is an operator satisfying our modified hypothesis with σ (H) ⊆ [0,∞], then f (H) = 0. P r o o f. Let ǫ ∈ (0, 1) . Define fǫ (x) := f (x+ ǫ) so that supp (fǫ) = [− (a+ ǫ) ,−ǫ]. We observe that for all n there are constants pn ≥ 0 such that ‖ dnf dxn − dnfǫ dxn ‖∞ ≤ pnǫ. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 237 Narinder S. Claire By Lemma 1.10 (iii), we have that fǫ (H) = 0, moreover, a further application of Lemma 1.10 implies that for large enough n we have ‖f (H) ‖ = ‖f (H) − fǫ (H) ‖ ≤ cn n∑ r=0 0∫ −(a+1) \| drf (x) dxr − drfǫ (x) dxr \|〈x〉r−1dx ≤ ǫcn n∑ r=0 pr 0∫ −(a+1) 〈x〉r−1dx = ǫkn,f hence our result. Corollary 6.9. If f and g are in A such that f \|[0,∞] = g\|[0,∞] and σ (H) ⊆ [0,∞], then f (H) − g (H) = 0. Theorem 6.10. If H satisfies our modified hypothesis with spectrum σ (H) ⊆ [0,∞), then there is a functional calculus γH : A+ → L (B) such that for all f ∈ A+ ∩A γH (f) = − 1 π ∫∫ C ∂f̃ ∂z (z −H)−1 dxdy. P r o o f. Let f+ ∈ A+, then by Seeley’s Extension Theorem there exists an extension f ∈ A. We define γH (f+) := f (H). This definition is independent of the particular extension by Corollary 6.9. The functional analytic properties are inherited from the extension. Theorem 6.11. (Refinement of Theorem 10 of [2]). Let n ≥ 1 be an integer and t > 0. If we denote the operator γH ( e−snt ) by e−Hnt, then e−Hn(t1+t2) = e−Hnt1e−Hnt2 for all n ≥ 1 and 0 < t ≤ 1. Acknowledgements. This research was funded by an EPSRC Ph.D grant 95-98 at Kings College, London. I am very grateful to E. Brian Davies for giving me this problem, his encouragement since and for continuing to be a mentor in Mathematics long after having finished supervising my Ph.D. I am indebted to Anita for all her support. I am immensely grateful to the referee for some very helpful comments and suggestions. 238 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 Spectral Mapping Theorem for the Davies–Helffer–Sjöstrand Functional Calculus References [1] B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper. Schrödinger Operators, H. Holden and A. Jensen (Eds.), Sønderborg, 1988; Lecture Notes in Phys., Vol. 345, Springer–Verlag, Berlin, 1989, 118–197. [2] E.B. Davies, The Functional Calculus. — J. London Math. Soc. 52 (1995), No. 1, 166–176. [3] E.M. Dyn’kin, An Operator Calculus Based upon the Cauchy–Green Formula, and the Quasi-Analyticity of the Classes D(h). — Sem. Math. V.A. Steklov Math. Inst. Leningrad 19 (1972), 128–131. [4] A. Bátkai and E. Fašanga, The Spectral Mapping Theorem for Davies’ Functional Calculus. — Rev. Roumaine Math. Pures Appl. 48 (2003), No. 4, 365–372. [5] E.B. Davies, Linear Operators and their Spectra. Cambridge Studies in Advanced Mathematics, 106, Cambridge Univ. Press, Cambridge, 2007. [6] E.B. Davies, Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, 42, Cambridge Univ. Press, Cambridge, 1995. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol 1. Springer, New York, 1993. [8] S.T. Seeley, Extensions of C∞ functions defined on a half space. — Proc. Amer. Math. Soc. 15 (1964), 625–626. Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 239

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