A remark on the H -calculus

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1 A remark on he H -calculus Nigel J. Kalon Absrac If A, B are secorial operaors on a Hilber space wih he same domain range, if Ax Bx A 1 x B 1 x, hen i is a resul of Auscher, McInosh Nahmod ha if A has an H calculus hen so does B. On an arbirary Banach space his is rue wih he addiional hypohesis on B ha i is almos R-secorial as was shown by he auhor, Kunsmann Weis in a recen preprin. We give an alernaive approach o his resul. MSC (2): 47A6. Received 17 Augus 26 / Acceped 21 Augus Inroducion In [1] he auhors showed ha if X is a Hilber space A, B are secorial operaors wih he same domain range saisfying esimaes Ax Bx x Dom (A) (1.1) A 1 x B 1 x x Ran (A) (1.2) hen if one of (A, B) admis an H calculus hen so does he oher. Resuls of his ype are useful in applicaions were sudied in [7] for arbirary Banach spaces. In ha paper, a similar resul (Theorem 5.1) is proved under he addiional hypohesis ha A is almos R-secorial. In his noe we give a raher differen approach o his resul. We replace he almos R-secorialiy assumpion by he echnically weaker assumpion of almos U-secorialiy, alhough his is probably no of grea significance. However, our approach here is perhaps a lile simpler. We also poin ou The auhor was suppored by NSF gran DMS

2 ha some addiional assumpion is necessary in arbirary Banach spaces; here are examples of secorial operaors A, B saisfying (1.1) (1.2) bu such ha only one has an H calculus. I is possible o consider esimaes on fracional powers our resuls can be exended in his direcion (as in [7]); however o keep he exposiion simple we will no discuss his poin. We also poin ou ha our approach is really based on an inerpolaion mehod, known as he Gusavsson-Peere mehod [5] (see also [4]); bu o avoid cerain echnicaliies we have no made his explici. 2 U-bounded collecions of operaors Le X be a complex Banach space. A family T of operaors T : X X is called U-bounded if here is a consan C such ha if (x j ) n X, (x j) n X, (T j ) n T, T j x j, x j C sup a j =1 a j x j sup a j =1 a j x j. The bes such consan C is called he U-bound for T is denoed U(T). This concep was inroduced in [8]. We recall ha T is called R-bounded if here is a consan C such ha if (x j ) n X, (T j ) n T, (E ɛ j T x j 2 ) 1/2 C(E ɛ j x j 2 ) 1/2. Here (ɛ j ) n is a sequence of independen Rademachers. The bes such consan C is called he R-bound for T is denoed R(T). An R-bounded family is auomaically U-bounded [8]. We will need he following elemenary propery: Proposiion 2.1. Suppose F : (, ) L(X) is a coninuous funcion ha T = {F () : < < } is U-bounded wih U-bound U(F ). Suppose g L 1 (R, d/). Then he family of operaors G(s) = g(s)f () d < s < is U-bounded wih consan a mos U(F ) g() d/. 82

3 Proof. Suppose (x j ) n X, (x j) n X wih sup a j =1 Then for s 1,..., s n R we have G(s j )x j, x j a j x j, sup a j =1 U(F ) a j x j 1. g() F (s 1 j )x j, x j d g() d. 3 Secorial operaors Le X be a complex Banach space le A be a closed operaor on X. A is called secorial if A has dense domain Dom (A) dense range Ran (A) = Dom (A 1 ) for some < ϕ < π he resolven (λ A) 1 is bounded for arg λ ϕ saisfies he esimae sup λ(λ A) 1 <. arg λ ϕ The infimum of such angles ϕ is denoed ω(a). Le Σ ϕ be he open secor {z : arg z < ϕ}. If f H (Σ ϕ ) we say ha f H (Σ ϕ ) if here exiss δ > such ha f(z) C max( z δ, z δ ). For f H (Σ ϕ ) where ϕ > ω(a) we can define f(a) by a conour inegral, which converges as a Bochner inegral in L(X). f(a) = 1 2πi f(λ)(λ A) 1 dλ where is he conour { e iνsgn : < < } ω(a) < ν < ϕ. We can hen esimae f(a) by f(a) C ϕ f(λ) dλ λ. 83

4 If we have a sronger esimae f(a) C f H (Σ ϕ) f H (Σ ϕ ) hen we say ha A has an H (Σ ϕ ) calculus; in his case we may exend he funcional calculus o define f(a) for every f H (Σ ϕ ). The infimum of all such angles ϕ is denoed by ω H (A). We will need a crierion for he exisence of an H -calculus. I will be convenien o use he noaion f λ (z) = f(λz) o le u(z) = z(1 + z) 2 so ha u H (Σ ϕ ) for all ϕ < π. The following crierion goes back o [2] [3]. A simple proof is given in [1]. Proposiion 3.1. Le A be a secorial operaor suppose < ϕ < π. Then he following are equivalen: (i) There is a consan C so ha u µ (A)x, x d C x x arg µ = ϕ, x X, x X. (ii) A has an H calculus wih ω H (A) π ϕ. Remark. (i) is equivalen by he Maximum Modulus Principle o u µ (A)x, x d C x x arg µ ϕ, x X, x X. If A is secorial we can define a closed operaor A on X by A x = x A wih domain Dom (A ) consising of all x such ha x x (Ax) exends o a bounded linear funcional on X. Then A need no be secorial since i need no have dense domain or range. Noe ha A x = (A ) 1 x = sup x, x x Dom (A ) A 1 x 1 x Ran (A) sup x, x x Ran (A ). Ax 1 x Dom (A) Thus if A B are secorial operaors saisfying (1.1) (1.2) hey will also saisfy Dom (A ) = Dom (B ), Ran (A ) = Ran (B ) A x B x x Dom (A ) (3.1) 84

5 (A ) 1 x (B ) 1 x x Ran (A ) (3.2) If A is a secorial operaor ϕ > ω(a) we shall ha f H (Σ ϕ ) is U-bounded (respecively R-bounded) for A if he family of operaors {f(a) : < < } is a U-bounded (respecively R-bounded) collecion. Proposiion 3.2. Suppose A has an H -calculus ha ϕ > ω H (A). Then for any f H (Σ ϕ ) we have ha f is R-bounded ( hus U-bounded) for A. Proof. Suppose ω(a) < ψ < ϕ. Then he map λ f(λa) is analyic on Σ ϕ ψ exends coninuously o he boundary. The operaors {f(2 k e ±i(ϕ ψ) A)} k Z are R-bounded (uniformly in < < ) by Theorem 3.3 of [8] he resul follows by Lemma 3.4 of he same paper. Suppose A is a secorial operaor on X ϕ > ω(a). We will say ha A is almos U-secorial (respecively almos R-secorial) if here is an angle ϕ such ha he se of operaors {λar(λ, A) 2 : arg λ ϕ} is U-bounded (respecively R-bounded). If we define u(z) = z(1+z) 2 his implies ha he funcions u λ (z) = u(λz) are uniformly U-bounded (respecively uniformly R- bounded) for arg λ π ϕ. The infimum of such angles is denoed ω U (A). By Lemma 3.4 of [8] his definiion is equivalen o ω U (A) = π sup{θ : u e ±iθ is U-bounded} or, respecively ω R (A) = π sup{θ : u e ±iθ is R-bounded}. Proposiion 3.3. Suppose A admis an H -calculus. Then A is almos R-secorial ( hence almos U-secorial) ω U (A) ω R (A) ω H (A). Proof. This follows from Proposiion 3.2. Lemma 3.1. Suppose A is almos U-secorial ϕ > ν > ω U (A). Then here is a consan C = C(ϕ) so ha if f H (Σ ϕ ) hen f is U-bounded for A wih U-bound U(f) C f(λ) dλ λ. 85

6 Proof. Fix ϕ > ψ > ν > ω U (A). We may wrie f(a) in he form f(a) = 1 f(λ)λ 1/2 A 1/2 (λ A) 1 dλ. 2πi Γ ψ Therefore he resul follows from Lemma 2.1 once we show ha he wo families of operaors {h(e ±iθ A) : < < } are U-bounded where θ = π ψ h(z) = z 1/2 (1 + z) 1. Consider g(z) = i log 1 + iz1/2 1 iz 1/2 π z 1 + z arg z < π. Then g H (Σ π ). Furhermore if Hence g e ±iθ g (z) = z 1/2 (1 + z) 1 π(1 + z) 2. H (Σ ψ ). For convenience we consider he case of +θ. Thus T = 1 g(e iθ λ)a(λ A) 2 dλ 2πi he family of operaors {T : < < } is U-bounded, again by Lemma 2.1. Now inegraion by pars shows ha T = eiθ ((e iθ λ) 1/2 (1 + e iθ λ) 1 π(1 + e iθ λ) 2 )λ(λ A) 1 dλ 2πi = 1 (h(e iθ λ) πu(e iθ λ))(λ A) 1 dλ 2πi = h(e iθ A) πu(e iθ A). Thus i follows ha he family {h(e iθ A) : < < } is U-bounded. 4 The main resuls If A is secorial hen he space Dom (A) Ran (A) is a Banach space (densely) embedded ino X under he norm Ax + A 1 x + x ; similarly Dom (A ) Ran (A ) is a Banach space embedded ino X under he norm A x + (A ) 1 x + x. 86

7 Theorem 4.1. Suppose A is a secorial operaor. In order ha A have an H -calculus wih ω H (A) = ϕ i is necessary sufficien ha: (i) A is almos U-secorial wih ω U (A) = ϕ. (ii) There exiss a consan C 1 so ha for each x X here is a coninuous funcion ξ : (, ) Dom (A) Ran (A) such ha N k= N a k 2 jk j A j ξ(2 k ) C 1 x, j = 1,, 1, a k 1, N = 1, 2,..., < < x, x = ξ(), x d x X. (iii) There exiss a consan C 2 so ha for each x X here is a coninuous funcion ξ : (, ) Dom (A ) Ran (A ) such ha N k= N a k 2 jk j (A j ) ξ (2 k ) C 2 x, j = 1,, 1, a k 1, N = 1, 2,..., < < x, x = x, ξ () d x X. Proof. Le us assume (i), (ii) (iii). Suppose θ < π ϕ x 1, x 1. Le ξ(), ξ () be chosen according o (ii) (iii). We define ξ() = Aξ() + 1 A 1 ξ() + 2ξ(), ξ () = A ξ () + 1 A ξ () + 2ξ (). Thus we have N a k 2 jk ξ(2 k ) 3C 1, j = 1,, 1, a k 1, N = 1, 2,..., < < k= N N a k 2 jk ξ (2 k ) 3C 2, j = 1,, 1, a k 1, N = 1, 2,..., < <. k= N Noe ha ξ : (, ) X ξ : (, ) X are boh coninuous ξ() = u(a) ξ() < < ξ () = (u(a)) ξ () < <. 87

8 If π arg µ > ν > ϕ we have For fixed r, s < u µ (ra)x, x > dr r = u µ (ra)ξ(s), ξ () d = = 2 1 j Z u µ (ra)ξ(s), ξ () d u µ (ra)ξ(s), ξ () d ds dr s r ds s u µ (ra)u(sa) ξ(s), (u(a)) ξ () d u rµ (2 j A)u s (2 j A)u(2 j A) ξ(s2 j ), ξ (2 j ) d 9C 1 C 2 U(u rµ u s u) C u(rµλ)u(sλ)u(λ) dλ λ, where C is consan independen of x, x. Inegraing over r, s gives: < u µ (ra)x, x > dr ( ) ( ) 2 r C. u µ (λ) dλ λ u(λ) dλ λ This esimae shows, by Proposiion 3.1, ha A has an H calculus wih ω H (A) ϕ. Since ω U (A) ω H (A) by Proposiion 3.3 we have equaliy. To complee he proof we show ha if A has an H calculus hen (i), (ii) (iii) hold ha ω U (A) ω H (A). To show (ii) (iii) we observe ha 12 (u(z)) 2 d = 1. Noe ha z j u(z) 2 H (Σ ϕ ) for j = 1,, 1. I follows easily ha if x X x X hen dr r ξ() = 12u(A) 2 x, ξ () = 12(u(A) 2 ) x give he required funcions. For (i) observe ha ω U (A) ω H (A) bu he firs par of he proof shows equaliy. 88

9 Theorem 4.2. Suppose A B are secorial operaors such ha Dom (A) = Dom (B), Ran (A) = Ran (B) for a suiable consan C we have C 1 Ax Bx C Ax x Dom (A) C 1 A 1 x B 1 x C A 1 x x Ran (A). Suppose A has an H calculus. Then he following are equivalen: (i) B has an H calculus wih ω H (B) = ϕ. (ii) B is almos U-secorial ω U (B) = ϕ. Proof. This is now immediae from Theorem 4.1 using (3.1) (3.2). If X is a Hilber space hen he assumpion ha B is almos U-secorial is redundan his reduces o he resul of Auscher, McInosh Nahmod [1]. However, in general his assumpion canno be eliminaed. I suffices o ake a secorial operaor A wih an H calculus wih ω H (A) > ω(a). Such examples exis [6]; in fac examples are known on subspaces of L p when 1 < p < 2 [9]. Now fix θ wih π ω H (A) < θ < π ω(a). Thus e ±iθ A are secorial wih ω(e ±iθ A) ω(a)+π θ. However if boh have an H calculus we would deduce ha for a suiable consan C u(e ±iθ A)x, x d C x x x X, x X which would imply ha ω H (A) π θ. This conradicion implies ha a leas one of e ±iθ A fails o have an H calculus. However if B = e ±iθ A hen (1.1) (1.2) are rivially saisfied. References [1] Auscher, P., McInosh, A., Nahmod, A., Holomorphic funcional calculi of operaors, quadraic esimaes inerpolaion, Indiana Univ. Mah. J. 46 (1997), [2] Boyadzhiev, K., delaubenfels, R., Semigroups resolvens of bounded variaion, imaginary powers H funcional calculus, Semigroup Forum, 45 no.3 (1992),

10 [3] Cowling, M., Dous, I., McInosh, A., Yagi, A., Banach space operaors wih a bounded H funcional calculus, J. Ausral. Mah. Soc. Ser. A 6 no.1 (1996), [4] Cwikel, M., Kalon, N. J., Inerpolaion of compac operaors by he mehods of Calderón Gusavsson-Peere, Proc. Edinburgh Mah. Soc. 38 no.2 (1995), [5] Gusavsson, J., Peere, J., Inerpolaion of Orlicz spaces, Sudia Mah. 6 no. 1 (1977), [6] Kalon, N. J., A remark on secorial operaors wih an H -calculus, in Trends in Banach spaces operaor heory, Memphis, TN, 21, Conemp. Mah. 321, Amer. Mah. Soc. (Providence, RI) 23, [7] Kalon, N. J., Kunsmann, P. C., Weis, L., Perurbaion Inerpolaion Theorems for he H -Calculus wih Applicaions o Differenial Operaors, Mah. Ann., o appear. [8] Kalon, N. J., Weis, L., The H -calculus sums of closed operaors, Mah. Ann. 321 (21), [9] Kalon, N. J., Weis, L., Euclidean srucures heir applicaions, in preparaion. [1] Kunsmann, P. C., Weis, L., Maximal L p -regulariy for parabolic equaions, Fourier muliplier heorems H -funcional calculus, in Funcional analyic mehods for evoluion equaions, Lecure Noes in Mah. 1855, Springer Verlag 24, Nigel Kalon, Deparmen of Mahemaics, Universiy of Missouri-Columbia, Columbia, MO nigel@mah.missouri.edu 9