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Abstract and Applied Analysis semigroups in the class of scalar type spectral operators
semigroups in the class of scalar type spectral operators
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Tập:
2004
Năm:
2004
Ngôn ngữ:
english
Tạp chí:
Abstract and Applied Analysis
DOI:
10.1155/s1085337504403054
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PDF, 1,84 MB
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A CHARACTERIZATION OF THE GENERATORS OF ANALYTIC C0 SEMIGROUPS IN THE CLASS OF SCALAR TYPE SPECTRAL OPERATORS MARAT V. MARKIN Received 30 January 2004 To my beloved grandmothers, Polina KhokhmovichRyklina and Berta KrasnovaRyklina In the class of scalar type spectral operators in a complex Banach space, a characterization of the generators of analytic C0 semigroups in terms of the analytic vectors of the operators is found. 1. Introduction Let A be a linear operator in a Banach space X with norm · , def C ∞ (A) = ∞ D An , (1.1) n =0 and 0 ≤ β < ∞. The sets of vectors def def Ᏹ{β} (A) = f ∈ C ∞ (A)  ∃α > 0, ∃c > 0 : An f ≤ cαn [n!]β , n = 0,1,... , Ᏹ(β) (A) = f ∈ C ∞ (A)  ∀α > 0 ∃c > 0 : An f ≤ cαn [n!]β , n = 0,1,... (1.2) are called the βthorder Gevrey classes of the operator A of Roumie’s and Beurling’s types, respectively. In particular, Ᏹ{1} (A) and Ᏹ(1) (A) are, correspondingly, the celebrated classes of analytic and entire vectors [6, 17]. Obviously, Ᏹ(1) (A) ⊆ Ᏹ{1} (A). (1.3) In [7, 8] and later in [19, 20], it was established that, for a selfadjoint nonpositive operator A in a complex Hilbert space H, Ᏹ(1) (A) = R etA , Ᏹ{1} (A) = t>0 Copyright © 2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:12 (2004) 1007–1018 2000 Mathematics Subject Classification: 47B40, 47D03, 47B15, 34G10 URL: http://dx.doi.org/10.1155/S1085337504403054 R etA , t>0 (1.4) 1008 One characterization of analytic semigroups where R(·) is the range of an operator, the exponentials understood in the sense of the operational calculus (o.c.) for normal operators etA := C etλ dEA (λ), t > 0, (1.5) EA (·) is the operator’s resolution of the identity (see, e.g., [3, 18]). In [9], it was proved that the second equality in (1.4) holds in a more general case, namely, when A generates an analytic C0 semigroup {etA  t ≥ 0} in a complex Banach space X. Later, in [12], it was demonstrated that, in the c; lass of normal operators in a complex Hilbert space, each of the equalities (1.4) characterizes the generators of the analytic semigroups. The purpose of the present paper is to stretch out the results of [12] to the case of scalar type spectral operators in a complex Banach space. It is absolutely fair of the reader to anticipate that abandoning the comforts of a Hilbert space would inevitably require introducing new approaches and techniques. 2. Preliminaries Henceforth, unless specified otherwise, A is a scalar type spectral operator in a complex Banach space X with norm · and EA (·) is its spectral measure (s.m.) (the resolution of the identity), the operator’s spectrum σ(A) being the support for the latter [1, 4]. Note that, in a Hilbert space, the scalar type spectral operators are those similar to the normal ones [21]. For such operators, there has been developed an o.c. for complexvalued Borel measurable functions on C [1, 4], F(·) being such a function, a new scalar type spectral operator, F(A) = C F(λ)dEA (λ), (2.1) is defined as follows: F(A) f := lim Fn (A) f , n→∞ f ∈ D F(A) , D F(A) := f ∈ X  lim Fn (A) f exists , (2.2) n→∞ D(·) is the domain of an operator, where Fn (·) := F(·)χ{λ∈CF(λ)≤n} (·), n = 1,2,..., (2.3) χα (·) is the characteristic function of a set α, and Fn (A) := C Fn (λ)dEA (λ), n = 1,2,..., (2.4) Marat V. Markin 1009 being the integrals of bounded Borel measurable functions on C, are bounded scalar type spectral operators on X defined in the same manner as for normal operators (see, e.g., [3, 18]). The properties of the s.m., EA (·), and the o.c. underlying the entire subsequent argument are exhaustively delineated in [1, 4]. We just observe here that, due to its strong countable additivity, the s.m. EA (·) is bounded, that is, there is an M > 0 such that, for any Borel set δ, EA (δ) ≤ M, (2.5) see [2]. Observe that, in (2.5), the notation · was used to designate the norm in the space of bounded linear operators on X. We will adhere to this rather common economy of symbols in what follows, adopting the same notation for the norm in the dual space X ∗ as well. With F(·) being an arbitrary complexvalued Borel measurable function on C, for any f ∈ D(F(A)), g ∗ ∈ X ∗ and arbitrary Borel sets δ ⊆ σ, we have (see [2]) σ F(λ) dv f ,g ∗ ,λ ≤ 4sup δ ⊆σ = 4sup δ ⊆σ = 4sup δ ⊆σ = 4sup δ ⊆σ δ σ F(λ)d EA (λ) f ,g ∗ χδ (λ)F(λ)d EA (λ) f ,g ∗ σ χδ (λ)F(λ)dEA (λ) f ,g ∗ (by the properties of the o.c.) (by the properties of the o.c.) (2.6) EA (δ)EA (σ)F(A) f ,g ∗ ≤ 4sup EA (δ)EA (σ)F(A) f g ∗ δ ⊆σ ≤ 4sup EA (δ) EA (σ)F(A) f g ∗ by (2.5) δ ⊆σ ≤ 4M EA (σ)F(A) f g ∗ . For the reader’s convenience, we reformulate here Proposition 3.1 of [14], heavily relied upon in what follows, which allows to characterize the domains of the Borel measurable functions of a scalar type spectral operator in terms of positive measures (see [14] for a complete proof). Proposition 2.1 [14]. Let A be a scalar type spectral operator in a complex Banach space X and let F(·) be a complexvalued Borel measurable function on C. Then, f ∈ D(F(A)) if and only if the following hold: (i) for any g ∗ ∈ X ∗ , C F(λ) dv f ,g ∗ ,λ < ∞, (2.7) 1010 One characterization of analytic semigroups (ii) sup {g ∗ ∈X ∗ g ∗ =1} {λ∈CF(λ)>n} F(λ) dv f ,g ∗ ,λ −→ 0 as n −→ ∞. (2.8) As was shown in [13], a scalar type spectral operator A in a complex Banach space X generates an analytic C0 semigroup, if and only if, for some real ω and 0 < θ ≤ π/2, σ(A) ⊆ λ ∈ C  arg(λ − ω) ≥ π +θ , 2 (2.9) where arg · is the principal value of the argument from the interval (−π,π] (see [15] for generalizations), in which case the semigroup consists of the exponentials etA = C etλ dEA (λ), t ≥ 0. (2.10) It is also to be noted that, according to [16], for a scalar type spectral operator A in a complex Banach space X, Ᏹ{1} (A) ⊇ D etA , Ᏹ(1) (A) ⊇ t>0 D etA , (2.11) t>0 the inclusions turning into equalities provided the space X is reflexive. 3. The principal statement Theorem 3.1. Let A be a scalar type spectral operator in a complex Banach space X. Then, each of equalities (1.4), the operator exponentials etA , t > 0, defined in the sense of the o.c. for scalar type spectral operators, is necessary and suﬃcient for A to be the generator of an analytic C0 semigroup. Proof Necessity. We consider the general of A being a generator of an analytic C0 semigroup {etA  t ≥ 0} in a complex Banach space X, without the assumption of A being a scalar type spectral operator. First, note that the inclusions Ᏹ{1} (A) ⊇ R etA , t>0 Ᏹ(1) (A) ⊇ R etA , t>0 (3.1) Marat V. Markin 1011 immediately follow from the estimate n n tA A e ≤ eωt M n!, tn n = 1,2,..., t > 0 (3.2) with some positive ω and M, known for analytic C0 semigroups (see, e.g., [11]). We show now that the inverse inclusions hold even in a more general case, when A generates a C0 semigroup {etA  t ≥ 0} not necessarily analytic. Let f be an analytic (entire) vector of the operator A, then, for some (any) δ > 0, the power series ∞ (−A)n f n! n =0 λn (3.3) converges whenever λ < δ. Formally designating the series by eλ(−A) f and diﬀerentiating it termwise, with the closedness of A in view, we obtain d λ(−A) f = −Aeλ(−A) f , e dλ eλ(−A) f ∈ D(A), λ < δ. (3.4) Considering that for any g ∈ D(A), d tA e g = AetA g = etA Ag, dt t ≥ 0, (3.5) (see [5, 10]), we have, for all 0 ≤ t < δ, d tA t(−A) d d f = eAs et(−A) f s=t + eAt et(−A) f e e dt ds dt = AetA et(−A) f + eAt − Aet(−A) f (3.6) = AeAt e−At f − AeAt e−At f = 0. This implies that, for all 0 ≤ t < δ, etA et(−A) f = eAs es(−A) f s=0 = f . (3.7) Therefore, Ᏹ {1} (A) ⊆ R e At Ᏹ (A) ⊆ (1) t>0 Suﬃciency. We prove this part by contrapositive. R e t>0 At . (3.8) 1012 One characterization of analytic semigroups As was noted in Section 2, for a scalar type spectral operator A, its being the generator of an analytic C0 semigroup is equivalent to inclusion (2.9) with some real ω and 0 < θ ≤ π/2. Hence, as is easily seen, the negation of the fact that A generates an analytic C0 semigroup implies that for any b > 0, the set σ(A) \ λ ∈ C  Reλ ≤ −b Im λ (3.9) is unbounded. In particular, for any natural n, the set σ(A) \ λ ∈ C  Reλ ≤ − 1  Imλ n2 (3.10) is unbounded. Hence, we can choose a sequence of points of the complex plane {λn }∞ n=1 in the following way: λn ∈ σ(A), n = 1,2,... ; 1 Reλn > − 2  Imλ, n = 1,2,... ; n λn > max n, λn−1 , n = 1,2,.... λ0 := 0, (3.11) The latter, in particular, implies that the points λn are distinct: λi = λ j , i = j. (3.12) Since the set λ ∈ C  Reλ > − 1  Imλ n2 (3.13) is open in C for any n = 1,2,..., there exists such an εn > 0 that this set contains together with the point λn the open disk centered at λn : ∆n = λ ∈ C  λ − λn < εn , (3.14) 1  Imλ, 2 n λ > max n, λn−1 . (3.15) that is, for any λ ∈ ∆n , Reλ > − Moreover, since the points λn are distinct, we can regard that the radii of the disks, εn , are chosen to be small enough so that 1 0 < εn < , n = 1,2,... ; n ∆i ∩ ∆ j = ∅, i = j (the disks are pairwise disjoint). (3.16) Marat V. Markin 1013 Note that, by the properties of the s.m., the latter implies that the subspaces EA (∆n )X, n = 1,2,..., are nontrivial, since ∆n ∩ σ(A) = ∅ and ∆n is open and EA ∆i EA ∆ j = 0, i = j. (3.17) Thus, choosing a unit vector en in each subspace EA (∆n )X, we obtain a vector sequence such that EA ∆i e j = δi j ei (3.18) (δi j is the Kronecker delta symbol). The latter, in particular, implies that the vectors {e1 ,e2 ,... } are linearly independent and that dn := dist en ,span ek  k ∈ N, k = n > 0, n = 1,2,.... (3.19) Furthermore, dn −→ 0 n −→ ∞. (3.20) Indeed, assuming the opposite, dn → 0 as n → ∞, would imply that, for any n = 1,2,..., there is an fn ∈ span({ek  k ∈ N, k = n}) such that en − fn < dn + 1/n, whence en = EA (∆n )(en − fn ) → 0, which is a contradiction. Therefore, there is a positive ε such that n = 1,2,.... dn ≥ ε, (3.21) As follows from the HahnBanach theorem, for each n = 1,2,..., there is an en∗ ∈ X ∗ such that ∗ e = 1, ei ,e∗j = δi j di . n (3.22) Let g ∗ := ∞ 1 n2 n =1 en∗ . (3.23) On one hand, for any n = 1,2,..., v en ,g ∗ ,∆n ≥ = EA ∆n en ,g ∗ en ,g ∗ ε ≥ 2. n = dn n2 by (3.18) by (3.21) (3.24) 1014 One characterization of analytic semigroups On the other hand, for any n = 1,2,..., v en ,g ∗ ,∆n (δ being an arbitrary Borel subset of ∆n , [2]) ≤ 4sup EA (δ) en g ∗ EA (δ)en ,g ∗ ≤ 4sup δ by (2.5) δ ≤ 4M g ∗ . (3.25) Concerning the sequence of the real parts, {Reλn }∞ n=1 , there are two possibilities: it is either bounded below, or not. We consider each of them separately. First, assume that the sequence {Reλn }∞ n=1 is bounded below, that is, there is such an ω > 0 that n = 1,2,.... Reλn ≥ ω, (3.26) Observe that this fact immediately implies that the operators e−tA , t > 0, are bounded and, thus, defined on the entire X [1, 4]. Therefore, R(etA ) = D(e−tA ) = X, t > 0. Let ∞ 1 f := n =1 n2 en . (3.27) As can be easily deduced from (3.17), EA ∆n f = EA 1 en , n2 ∞ n = 1,2,..., (3.28) ∆ f = f. n =1 n For an arbitrary t > 0, we have C etλ dv f ,g ∗ ,λ = = = ≥ ≥ by (3.28); = C etλ dv EA ∞ n=1 ∆n ∞ ∞ 1 n2 n =1 n =1 ∆n etλ dv en ,g ∗ ,λ etn v f ,g ∗ ,∆n ∞ εetn n =1 etλ dv EA ∆n f ,g ∗ ,λ n2 n =1 ∞ 1 ∆n f ,g ∗ ,λ etλ dv EA ∆n f ,g ∗ ,λ n=1 ∆n ∞ n4 = ∞. (by the properties of the o.c.) by (3.28) for λ ∈ ∆n , by (3.15), λ ≥ n by (3.24) (3.29) Marat V. Markin 1015 This, by [14, Proposition 3.1], implies that f∈ D etA . (3.30) t>0 Then, by (2.11), moreover, f ∈ Ᏹ{1} (A). (3.31) Therefore, equalities (1.4) do not hold. Now, suppose that the sequence {Reλn }∞ n=1 is unbounded below, that is, there is a sub(k ≤ n(k)) such that sequence {Reλn(k) }∞ k =1 Reλn(k) −→ −∞ as k −→ ∞. (3.32) Without the loss of generality, we can regard that Reλn(k) ≤ −k, k = 1,2,.... (3.33) Let f := ∞ ek Reλn(k) en(k) . (3.34) k =1 Similarly to (3.17), we have EA ∆n(k) f = ek Reλn(k) en(k) , EA ∞ n = 1,2,..., (3.35) ∆n(k) f = f . n =1 For any t > 0 and an arbitrary g ∗ ∈ X ∗ , C e−t Reλ dv f ,g ∗ ,λ = = = ∞ k =1 ∆n(k) ∞ k=1 ∆n(k) ∞ k =1 ≤ ∞ e−t Re λ dv f ,g ∗ ,λ (by the properties of the o.c.) etλ dv EA ∆n(k) f ,g ∗ ,λ ek Reλn(k) ∆n(k) e−t Reλ dv en(k) ,g ∗ ,λ by (3.35) ek Reλn(k) et(− Reλn(k) +1) v en(k) ,g ∗ ,∆n(k) k =1 ∞ e(k−t)Reλn(k) < ∞. ≤ 4M g ∗ et k =1 by (3.16) by (3.25) (3.36) 1016 One characterization of analytic semigroups Indeed, for λ ∈ ∆n(k) , by (3.16), − Reλ = − Reλn(k) + (Reλn(k) − Re λ) ≤ − Re λn(k) + λn(k) − λ ≤ − Reλn(k) + εn(k) ≤ − Reλn(k) + 1 and for all natural k’s large enough so that k − t ≥ 1, due to (3.33), e(k−t)Reλn(k) ≤ e−k . (3.37) Similarly, for any t > 0, sup {g ∗ ∈X ∗ g ∗ =1} {λ∈Ce−t Reλ >n} = sup {g ∗ ∈X ∗ g ∗ =1} ≤ et ∞ ∞ et e(k−t)Reλn(k) ≤ et ≤ 4Met {λ∈∆n(k) e−t Re λ >n} {g ∗ ∈X ∗ g ∗ =1} {g ∗ ∈X ∗ g ∗ =1} e−t Reλ dv en(k) ,g ∗ ,λ v f ,g ∗ , λ ∈ ∆n(k)  e−t Reλ > n sup k =1 ∞ sup e(k−t)Reλn(k) ek Reλn(k) k =1 k =1 ∞ e−t Reλ dv f ,g ∗ ,λ 4M EA λ ∈ ∆n(k)  et Reλ > n e(k−t)Reλn(k) EA λ ∈ C  e−t Re λ > n by (2.6) ∗ f g f k =1 (by the strong continuity of the s.m. −→ 0 as n −→ ∞). (3.38) According to [14, Proposition 3.1], (3.36) and (3.38) imply that f ∈ D e−tA = t>0 R etA . (3.39) t>0 However, for an arbitrary t > 0, we have C etλ dv f ,g ∗ ,λ = ∞ ek Reλn(k) k =1 ≥ ∞ ∆n(k) etλ dv en(k) ,g ∗ ,λ by the properties of the o.c. and (3.35) ek Reλn(k) e−tn(k) (Reλn(k) +1) dv en(k) ,g ∗ ,∆n(k) 2 by (3.15) and (3.16) k =1 = ∞ e−tn(k) e(tn(k) −k)(− Reλn(k) ) dv en(k) ,g ∗ ,∆n(k) 2 2 by (3.24) k =1 ≥ ∞ k =1 e−tn(k) e(tn(k) −k)(− Re λn(k) ) 2 2 ε = ∞. n(k)2 (3.40) Marat V. Markin 1017 Indeed, for λ ∈ ∆n(k) , by (3.15) and (3.16), λ ≥  Imλ ≥ −n(k)2 Re λ ≥ −n(k)2 (Re λn(k) +  Reλ − Reλn(k) ) ≥ −n(k)2 (Reλn(k) + 1), and for all natural k’s large enough so that tn(k)2 − k > 0, due to (3.33), we have etn(k) −tn(k) −kn(k) ε ≥ε −→ ∞, 2 n(k) n(k)2 3 e−tn(k) e(tn(k) −k)(− Reλn(k) ) 2 2 2 as k −→ ∞. (3.41) Whence, by [14, Proposition 3.1], we infer that f ∈ t>0 D etA . Then, by (2.11), moreover f ∈ Ᏹ{1} (A). Therefore, equalities (1.4) do not hold in this case either. With all the possibilities concerning {Reλn }∞ n=1 having been analyzed, we conclude that the suﬃciency part has been proved by contrapositive. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217– 274. N. Dunford and J. T. Schwartz, Linear Operators. Part I. 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L. van Eijndhoven and J. de Graaf, Trajectory Spaces, Generalized Functions and Unbounded Operators, Lecture Notes in Mathematics, vol. 1162, SpringerVerlag, Berlin, 1985. J. Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355–361. Marat V. 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