On the Carleman Classes of Vectors of a Scalar Type Spectral Operator

IJMMS 2004:60, 3219–3235 PII. S0161171204311117 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON THE CARLEMAN CLASSES OF VECTORS OF A SCALAR TYPE SPECTRAL OPERATOR

MARAT V. MARKIN

Received 6 November 2003

To my teachers, Drs. Miroslav L. Gorbachuk and Valentina I. Gorbachuk

The Carleman classes of a scalar type spectral operator in a reﬂexive Banach space are characterized in terms of the operator’s resolution of the identity. A theorem of the Paley- Wiener type is considered as an application.

2000 Mathematics Subject Classiﬁcation: 47B40, 47B15, 47B25, 30D60.

1. Introduction. As was shown in [8](seealso[9, 10]), under certain conditions, the Carleman classes of vectors of a normal operator in a complex Hilbert space can be characterized in terms of the operator’s spectral measure (the resolution of the identity). The purpose of the present paper is to generalize this characterization to the case of a scalar type spectral operator in a complex reﬂexive Banach space.

2. Preliminaries

2.1. The Carleman classes of vectors. Let A be a linear operator in a Banach space · { }∞ X with norm , mn n=0 a sequence of positive numbers, and ∞ def C∞(A) = D An (2.1) n=0

(D(·) is the domain of an operator). The sets def= ∈ ∞ |∃ ∃ n ≤ n = C{mn}(A) f C (A) α>0, c>0: A f cα mn,n 0,1,2,... , (2.2) def= ∈ ∞ |∀ ∃ n ≤ n = C(mn)(A) f C (A) α>0 c>0: A f cα mn,n 0,1,2,... are called the Carleman classes of vectors of the operator A corresponding to the se- { }∞ quence mn n=0 of Roumie’s and Beurling’s types, respectively. Obviously, the inclusion

⊆ C(mn)(A) C{mn}(A) (2.3) holds. β βn For mn := [n!] (or, due to Stirling’s formula,formn := n ), n = 0,1,2,... (0 ≤ β< ∞), we obtain the well-known βth-order Gevrey classes of vectors, Ᏹ{β}(A) and Ᏹ(β)(A), 3220 MARAT V. MARKIN respectively. In particular, Ᏹ{1}(A) are the analytic and Ᏹ(1)(A) are the entire vectors of the operator A [7, 17]. { }∞ The sequence mn n=0 will be subject to the following condition. (WGR) For any α>0, there exist such a C = C(α) > 0 that

n Cα ≤ mn,n= 0,1,2,.... (2.4)

Note that the name WGR originates from the words “weak growth.” Under this condition, the numerical function

∞ λn T(λ):= m , 0 ≤ λ<∞, 00 := 1 , (2.5) 0 m n=0 n

ﬁrst introduced by Mandelbrojt [15], is well deﬁned. This function is nonnegative, continuous,andincreasing. As established in [8](seealso[9, 10]), for a normal operator A with a spectral measure · · · { }∞ EA( ) in a complex Hilbert space H with inner product ( , ) and the sequence mn n=0 satisfying the condition (WGR),

= | | C{mn}(A) D T(t A ) , t>0 (2.6) = | | C(mn)(A) D T(t A ) , t>0 the normal operators T(t|A|) (0

T(t|A|) := T(t|λ|)dEA, σ(A) (2.7) 2 D T(t|A|) := f ∈ H | T (t|λ|) dEA(λ)f ,f < ∞ , σ(A) where the function T(·) can be replaced by any nonnegative, continuous, and increasing function L(·) deﬁned on [0,∞) such that

c1L γ1λ ≤ T(λ)≤ c2L γ2λ ,λ>R, (2.8)

with some positive γ1, γ2, c1, c2, and a nonnegative R. In particular, T(·) in (2.6) is replaceable by

λn S(λ) := m0 sup , 0 ≤ λ<∞, (2.9) n≥0 mn THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3221 or

∞ 1/2 2n = λ ≤ ∞ P(λ): m0 2 , 0 λ< , (2.10) n=0 mn

(see [10]).

2.2. Carleman ultradifferentiability. Let I be an interval of the real axis, C∞(I) the { }∞ set of all complex-valued functions strongly infinite differentiable on I, and mn n=0 a sequence of positive numbers.

  · ∈ ∞ |∀ ⊆ ∃ ∃  f( ) C (I) [a,b] I, α>0, c>0:   (n) ≤ n = def max a≤x≤b f (x) cα mn,n 0,1,2,... , C{ }(I) = (2.11) mn  ∞  f(·) ∈ C (I) |∀[a,b] ⊆ I, ∀α>0, ∃c>0:   (n) n maxa≤x≤b f (x) ≤ cα mn,n= 0,1,2,... are the Carleman classes of ultradiﬀerentiable functions of Roumie’s and Beurling’s types, respectively, [1, 12, 13, 14]. β βn In particular, for mn := [n!] (or, due to Stirling’s formula,formn := n ), n = 0,1,2,... (0 ≤ β<∞), these are the well-known βth-order Gevrey classes, Ᏹ{β}(I) and Ᏹ(β)(I), respectively, [6, 12, 13, 14]. Observe that Ᏹ{1}(I) is the class of the real analytic on I functions and Ᏹ(1)(I) is the class of entire functions, that is, the restrictions to I of analytic and entire functions, correspondingly, [15].

Note that condition (WGR), in particular, implies that limn→∞ mn =∞.Since,asis easily seen, the Carleman classes of vectors and functions coincide for the sequence { }∞ { }∞ mn n=1 and the sequence dmn n=1 for any d>0, without loss of generality, we can regard that

inf mn ≥ 1. (2.12) n≥0

2.3. Scalar type spectral operators. 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 (the resolution of the identity), the operator’s spectrum σ(A)being the support for the latter [2, 5]. 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 operational calculus for Borel measurable functions on C (on σ(A))[2, 5], F(·) being such a function; a new scalar type spectral operator

F(A)= F(λ)dEA(λ) = F(λ)dEA(λ) (2.13) C σ(A) 3222 MARAT V. MARKIN is deﬁned as follows:

F(A)f := lim Fn(A)f , f ∈ D F(A) , n→∞ (2.14) D F(A) := f ∈ X | lim Fn(A)f exists n→∞

(D(·) is the domain of an operator), where

Fn(·) := F(·)χ{λ∈σ(A)||F(λ)|≤n}(·), n = 1,2,..., (2.15)

(χα(·) is the characteristic function of a set α), and

Fn(A) := Fn(λ)dEA(λ), n = 1,2,..., (2.16) σ(A) being the integrals of bounded Borel measurable functions on σ(A),arebounded scalar type spectral operators on X deﬁned in the same manner as for normal operators (see, e.g., [4, 19]).

The properties of the spectral measure, EA(·), and the operational calculus underlying the entire subsequent argument are exhaustively delineated in [2, 5]. We just observe here that, due to its strong countable additivity, the spectral measure EA(·) is bounded [3], that is, there is an M>0 such that, for any Borel set δ,

EA(δ) ≤ M. (2.17)

Observe that, in (2.17), 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. Due to (2.17), for any f ∈ X and g∗ ∈ X∗ (X∗ is the dual space), the total variation ∗ ∗ v(f,g ,·) of the complex-valued measure EA(·)f ,g ( ·,· is the pairing between the space X and its dual, X∗)isbounded. Indeed, δ being an arbitrary Borel subset of σ(A),[3],

v f,g∗,σ (A) ∗ ∗ ≤ 4sup EA(δ)f ,g ≤ 4sup EA(δ) f g (by (2.17)) ⊆ ⊆ (2.18) δ σ(A) δ σ(A) ∗ ≤ 4Mf g .

For the reader’s convenience, we reformulate here [16, Proposition 3.1], heavily relied upon in what follows, which allows to characterize the domains of the Borel measurable THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3223 functions of a scalar type spectral operator in terms of positive measures (see [16]for a complete proof). On account of compactness, the terms spectral measure and operational calculus for scalar type spectral operators, frequently referred to, will be abbreviated to s.m. and o.c., respectively.

Proposition 2.1. Let A be a scalar type spectral operator in a complex Banach space X and F(·) a complex-valued Borel measurable function on C (on σ(A)). Then f ∈ D(F(A)) if and only if (i) for any g∗ ∈ X∗, |F(λ)|dv f,g∗,λ < ∞, (2.19) σ(A)

(ii) sup |F(λ)|dv f,g∗,λ → 0 as n →∞. (2.20) {g∗∈X∗|g∗=1} {λ∈σ(A)||F(λ)|>n}

Observe that, for F(·) being an arbitrary Borel measurable function on C (on σ(A)), for any f ∈ D(F(A)), g∗ ∈ X∗, and arbitrary Borel sets δ ⊆ σ, |F(λ)|dv f,g∗,λ (see [3]) σ ∗ ≤ 4sup F(λ)d EA(λ)f ,g δ⊆σ δ ∗ = 4sup χδ(λ)F(λ)d EA(λ)f ,g (by the properties of the o.c.) ⊆ δ σ σ ∗ = 4sup χδ(λ)F(λ)dEA(λ)f ,g (by the properties of the o.c.) ⊆ δ σ σ ∗ = 4sup EA(δ)EA(σ )F(A)f ,g δ⊆σ ∗ ≤ 4sup EA(δ)EA(σ )F(A)f g δ⊆σ ∗ ≤ 4sup EA(δ) EA(σ )F(A)f g (by (2.17)) δ⊆σ ∗ ∗ ≤ 4M EA(σ )F(A)f g ≤ 4M EA(σ ) F(A)f g . (2.21)

In particular, |F(λ)|dv f,g∗,λ (by (2.21)) σ(A) ∗ ≤ 4M E σ(A) F(A)f g A (2.22) (since EA(σ (A)) = I (I is the identity operator in X)) ∗ ≤ 4M F(A)f g . 3224 MARAT V. MARKIN

3. The Carleman classes of a scalar type spectral operator

Theorem 3.1. Let A be a scalar type spectral operator in a complex reﬂexive Banach { }∞ space X. If a sequence of positive numbers mn n=0 satisﬁes condition (WGR), equalities (2.6) hold, the scalar type spectral operators T(t|A|)(0 R, (3.1) with some positive γ1, γ2, c1, c2, and a nonnegative R.

Proof. First, we prove the replaceability of T(·) in (2.6)byanonnegative, continuous, and increasing function satisfying (3.1) with some positive γ1, γ2, c1, c2,anda nonnegative R ≥ 0. Let f ∈ T(t|A|) T(t|A|) . (3.2) t>0 t>0

Then, for some (any) 0

For any g∗ ∈ X∗, ∗ L γ1t|λ| dv f,g ,λ < ∞. (3.4) σ(A)

Indeed, ∗ L γ1t|λ| dv f,g ,λ σ(A) ∗ ∗ = L γ1t|λ| dv f,g ,λ + L γ1t|λ| dv f,g ,λ {λ∈σ(A)|t|λ|≤R} {λ∈σ(A)|t|λ|>R} ∗ ∗ ≤ L γ1R v f,g ,σ (A) + L γ1t|λ| dv f,g ,λ (by (2.18)) {λ∈σ(A)|t|λ|>R} ∗ ∗ ≤ L γ1R 4Mf g + L γ1t|λ| dv f,g ,λ (by (3.1)) {λ∈σ(A)|t|λ|>R} ∗ 1 ∗ ≤ L γ1R 4Mf g + F(t|λ|)dv f,g ,λ c1 {λ∈σ(A)|t|λ|>R} ∗ 1 ∗ ≤ L γ1R 4Mf g + F(t|λ|)dv f,g ,λ (by (3.3)) c1 σ(A) < ∞. (3.5) THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3225

Further,

∗ sup L γ1t|λ| dv f,g ,λ = 0 (3.6) ∗ ∗ ∗ {g ∈X |g =1} {λ∈σ(A)|t|λ|≤R, L(γ1t|λ|)>n}

for all suﬃciently large natural n’s since, when t|λ|≤R, L(γ1t|λ|) ≤ L(γ1R). On the other hand,

∗ L γ1t|λ| dv f,g ,λ (by (3.1)) { ∈ | | | | | } λ σ(A) t λ >R, L(γ1t λ )>n 1 ≤ T(t|λ|)dv f,g∗,λ (by (2.21)) c1 {λ∈σ(A)|t|λ|>R, T (t|λ|)>c1n} (3.7) 1 ∗ ≤ EA λ ∈ σ(A)| T(t|λ|)>c1n T(t|A|)f g c1 (by the continuity of the s.m.) → 0asn →∞.

Therefore, by Proposition 2.1, f ∈ D(L(γ1t|A|)). Thus, we have proved the inclusions

D T(t|A|) ⊆ D L(t|A|) , t>0 t>0 (3.8) D T(t|A|) ⊆ D L(t|A|) . t>0 t>0

Similarly, one can derive from (3.1) the inverse inclusions:

D T(t|A|) ⊇ D L(t|A|) , t>0 t>0 (3.9) D T(t|A|) ⊇ D L(t|A|) . t>0 t>0

Thus,

D T(t|A|) = D L(t|A|) , t>0 t>0 (3.10) D T(t|A|) = D L(t|A|) . t>0 t>0 3226 MARAT V. MARKIN

∈ ∈ ∞ Let f C{mn}(A) (C(mn)(A)). Then f C (A) and, for a certain (an arbitrary) α>0, there is a c>0 such that

n n A f ≤ cα mn,n= 0,1,2,.... (3.11)

For any g∗ ∈ X∗,

∞ 1 |λ|n T |λ| dv f,g∗,λ = dv f,g∗,λ 2α 2nαnm σ(A) σ(A)n=0 n (by the monotone convergence theorem)

∞ |λ|n = dv f,g∗,λ 2nαnm n=0 σ(A) n ∞ 1 = |λ|n dv f,g∗,λ (by (2.22)) 2nαnm n=0 n σ(A) ∞ 1 ∗ ≤ 4M Anf g (by (3.11)) 2nαnm n=0 n ∞ 1 ∗ ∗ ≤ 4Mc g = 8Mc g < ∞. 2n n=0 (3.12)

Let

∆n := λ ∈ σ(A)||λ|≤n ,n= 0,1,2,.... (3.13)

By the properties of the o.c., T((1/2α)|A|)EA(∆n), n = 0,1,2,..., is a bounded operator on X and

∞ k 1 n T |A| E ∆ ≤ 4M 2α A n 2kαkm k=0 k nk by condition (WGR), there is a C = C(α,n)> 0: ≤ C, k = 0,1,... (3.14) k α mk ∞ 1 ≤ 4MC = 8MC. 2k k=0 THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3227

For any 1 ≤ m

1 1 ∗ T |A| E ∆ f −T |A| E ∆ f,g 2α A n 2α A m (by the properties of the o.c.) 1 ∗ T |λ| dEA(λ)f ,g {λ∈σ(A)|m<|λ|≤n} 2α (by the properties of the o.c.) (3.15) 1 ∗ = T |λ| d EA(λ)f ,g {λ∈σ(A)|m<|λ|≤n} 2α 1 ≤ T |λ| dv f,g∗,λ (by (3.12)) {λ∈σ(A)|m<|λ|} 2α → 0asm →∞.

Since a reﬂexive Banach space is weakly complete (see, e.g., [3]), we infer that the se- { | | ∆ }∞ quence T((1/2α) A )EA( n)f n=1 weakly converges in X. This, considering the fact that, by the continuity of the s.m.,

EA ∆n f → f as n →∞ (3.16) and the closedness of the operator T((1/2α)|A|),implies

1 f ∈ D T |A| . (3.17) 2α

Therefore,

f ∈ D T(t|A|) D T(t|A|) , resp. , (3.18) t>0 t>0 which proves the inclusions

⊆ | | C{mn}(A) D T(t A ) , t>0 (3.19) ⊆ | | C(mn)(A) D T(t A ) . t>0

Now, we are to prove the inverse inclusions. 3228 MARAT V. MARKIN

Let

f ∈ D T(t|A|) D T(t|A|) . (3.20) t>0 t>0

Then, for a certain (any) t>0, f ∈ D(T(t|A|)). We infer from the latter that f ∈ C∞(A). Indeed, for an arbitrary N = 0,1,2,... and any g∗ ∈ X∗,

∞ tN [t|λ|]k |λ|N dv f,g∗,λ ≤ dv f,g∗,λ σ(A) mN σ(A) mk k=0 ∗ = T(t|λ|)dv f,g ,λ (3.21) σ(A) (by Proposition 2.1), < ∞.

Further, for any N = 0,1,2,...,

tN sup |λ|N dv f,g∗,λ { ∗∈ ∗| ∗= } {λ∈σ(A)|(tN /m )|λ|N >n} m g X g 1 N N ≤ sup T(t|λ|)dv f,g∗,λ (by Proposition 2.1), {g∗∈X∗|g∗=1} {λ∈σ(A)|T(t|λ|)>n} → 0asn →∞. (3.22)

By Proposition 2.1,(3.21)and(3.22)implythat

f ∈ C∞(A). (3.23)

Further, by (2.22),

sup T(t|λ|)dv f,g∗,λ (by (2.22)) { ∗∈ ∗| ∗= } σ(A) g X g 1 (3.24) ≤ 4M T(t|A|)f < ∞.

By (2.22),

Whence, for any n = 0,1,2,..., tn c ≥ sup |λ|n dv f,g∗,λ { ∗∈ ∗| ∗= } σ(A) mn g X g 1 n t n ∗ ≥ sup λ d EA(λ)f ,g mn {g∗∈X∗|g∗=1} σ(A) (by the properties of the o.c.) n t n ∗ ≥ sup λ dEA(λ)f ,g mn {g∗∈X∗|g∗=1} σ(A) (3.26) (by the properties of the o.c.) n t ∗ = sup Anf,g mn {g∗∈X∗|g∗=1} (as follows from the Hahn-Banach theorem) n t = Anf . mn Thus, for some (any) t>0, n 1 Anf ≤ c m ,n= 0,1,2,.... (3.27) t n Hence, ∈ f C{mn}(A) C(mn)(A), resp. , (3.28) which proves the inverse inclusions ⊇ | | C{mn}(A) D T(t A ) , t>0 (3.29) ⊇ | | C(mn)(A) D T(t A ) . t>0

From (3.19)and(3.29), we infer equalities (2.6).

Remark 3.2. Observe that the assumption of the reﬂexivity of the space X was utilized for proving the inclusions ⊆ | | C{mn}(A) D T(t A ) , t>0 (3.30) ⊆ | | C(mn)(A) D T(t A ) t>0 only. The inverse inclusions ⊇ | | C{mn}(A) D T(t A ) , t>0 (3.31) ⊇ | | C(mn)(A) D T(t A ) t>0 hold regardless whether X is reﬂexive or not. 3230 MARAT V. MARKIN

4. The Gevrey classes of a scalar type spectral operator. Let 0 <β<∞.Asiseasily β seen, the sequence mn = [n!] , n = 0,1,2,..., satisﬁes condition (WGR) and, thus, the function

∞ λn T(λ):= , 0 ≤ λ<∞, (4.1) [n!]β n=0 is well deﬁned. According to Stirling’s formula,

nβn ∼ (2πn)−β/2eβn[n!]β as n →∞. (4.2)

Hence, there is such a C = C(β) ≥ 1 such that

[n!]β ≤ nβn ≤ C(2πn)−β/2eβn[n!]β ≤ Ceβn[n!]β,n= 0,1,2,.... (4.3)

Taking this into account, we infer ∞ ∞ n ∞ n λn λn eβλ 1 2eβλ sup ≤ ≤ T(λ)≤ C = C ≥ nβn nβn nβn 2n nβn n 0 n=0 n=0 n=0 n ∞ n (4.4) 2eβλ 1 2eβλ ≤ C sup = 2C sup , 0 ≤ λ<∞. ≥ nβn 2n ≥ nβn n 0 n=0 n 0

Now, we consider the family of functions

λx ρ (x) := , 0 ≤ x<∞, 1 ≤ λ<∞ 00 := 1 . (4.5) λ xβx

It is easy to make sure that the function ρλ(·) attains its maximum value on [0,∞) at −1 1/β the point xλ = e λ . Therefore,

n x λ λ −1 1/β sup ≤ sup = ρ x = eβe λ . (4.6) βn βx λ λ n≥0 n x≥0 x

β −1 1/β For λ ≥ e ,letN be the integer part of xλ = e λ . Hence, N ≥ 1 and

λn λN sup ≥ = exp N lnλ−βN lnN ≥ nβn NβN n 0 (4.7) 1 −1 1/β ≥ exp x −1 lnλ−βx lnx = eβe λ ,λ≥ eβ. λ λ λ λ

Obviously, for all suﬃciently large positive λ’s,

−1 1/β 1 e−(βe /2)λ ≤ . (4.8) λ THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3231

Based on (4.4), (4.6), (4.7), and (4.8), for all suﬃciently large positive λ’s, n − β (ββ(e β/2β)λ)1/β 2e λ e ≤ T(λ)≤ 2C sup ≤ 2C supρ β (x) βn 2e λ n≥0 n x≥0 (4.9) −1 β 1/β β 1/β = 2Ceβe (2e λ) ≤ e(4β λ) .

Thus, by Theorem 3.1, in the considered case, the function T(λ) can be replaced by 1/β eλ (0 ≤ λ<∞) and we arrive at the following.

Corollary 4.1. Let A be a scalar type spectral operator in a complex reﬂexive Ba- nach space and 0 <β<∞.Then 1/β Ᏹ{β}(A) = D et|A| , t>0 (4.10) 1/β Ᏹ(β)(A) = D et|A| . t>0

In particular, for β = 1, Corollary 4.1 gives the description of the analytic and entire vectors of the scalar type spectral operator A. Corollary 4.1 generalizes the corresponding result of [8](seealso[9, 10]) for a normal operator in a complex Hilbert space. Observe that the inclusions 1/β Ᏹ{β}(A) ⊇ D et|A| , t>0 (4.11) 1/β Ᏹ(β)(A) ⊇ D et|A| . t>0 are valid without the assumption of the reﬂexivity of X (see Remark 3.2).

5. A theorem of the Paley-Wiener type. Consider the self-adjoint diﬀerential operator A = i(d/dx) (i is the imaginary unit) in the complex Hilbert space L2(−∞,∞). With the unitary equivalence of this operator and the operator of multiplication by the independent variable x in view, by Theorem 3.1 as well as by [9, 10], we arrive at the following theorem of the Paley-Wiener type [18, 22]. Theorem 5.1. { }∞ Let mn n=0 be a sequence of positive numbers satisfying condition (WGR), then ∞ ∈ ⇐⇒ ˆ 2 2 | | ∞ f C{mn}(A) C(mn)(A) f(x) T (t x )dx< (5.1) −∞

(fˆ is the Fourier transform of f ) for some (any) 0

[0,∞) and satisfying (3.1) with some positive γ1, γ2, c1, c2, and a nonnegative R.

The only natural question to be answered now is how the abstract smoothness rela- tive to the diﬀerential operator A in L2(−∞,∞) reveals itself as the smoothness in the ordinary sense. 3232 MARAT V. MARKIN

∈ n n = n For any f W2 (I), where I is an interval of the real axis and W2 (I) H (I) is the nth-order Sobolev space [20], let f(·) be the representative of the equivalence class f continuously differentiable n−1 times and such that f (n−1)(·) is absolutely continuous on I. For ∞ ∈ ∞ −∞ ∞ = n −∞ ∞ f W2 ( , ) : W2 ( , ), (5.2) n=0 let f(·) be the infinite-differentiable representative of the equivalence class f such that

∞ 2 f (n)(t) dt < ∞,n= 0,1,2,.... (5.3) −∞

Let ˆ −∞ ∞ def= ∈ ∞ −∞ ∞ |∀ ⊆ −∞ ∞ ∃ C{mn}( , ) f W2 ( , ) [a,b] ( , ) α>0, (n) n ∃c>0: max f (t) ≤ cα mn,n= 0,1,2,... , a≤t≤b (5.4) ˆ −∞ ∞ def= ∈ ∞ −∞ ∞ |∀ ⊆ −∞ ∞ ∀ C(mn)( , ) f W2 ( , ) [a,b] ( , ), α>0 (n) n ∃c>0: max f (t) ≤ cα mn,n= 0,1,2,... . a≤t≤b

{ }∞ We will impose upon the sequence mn n=0 an additional condition. (DI) There are an L>0andaγ>1 such that

n mn+1 ≤ Lγ mn,n= 0,1,2,....

Note that the name (DI) originates from the words “differentiation invariant” since, −∞ ∞ as is easily verifiable, under this condition, the Carleman classes C{mn}( , ) and −∞ ∞ · · C(mn)( , ) along with a function f( ) contain its first derivative, f ( ). β Observe that, for 0 ≤ β<∞, the Gevrey sequence mn = [n!] , n = 0,1,2,..., meets β condition (DI) with any γ>1. Indeed, in this case, mn+1/mn = (n+1) , n = 0,1,2,.... Lemma 5.2. { }∞ Let a sequence of positive numbers mn n=0 satisfy condition (DI).Then

⊆ ˆ −∞ ∞ C{mn}(A) C{mn}( , ), (5.5) ⊆ ˆ −∞ ∞ C(mn)(A) C(mn)( , ).

Proof. ∈ Let f C{mn}(A) (C(mn)(A)), Then

∈ ∞ −∞ ∞ f W2 ( , ), (5.6) and for some (any) α>0, there is a c>0 such that ∞ 1/2 = (n) 2 ≤ n = f L2(−∞,∞) f (x) dx cα mn,n0,1,2,.... (5.7) −∞ THE CARLEMAN CLASSES OF A SCALAR OPERATOR 3233

We ﬁx a ﬁnite segment [a,b] of the real axis. Then, according to the Sobolev embedding 1 theorems [20] (see also [22, 23]), the space W2 (a,b) is continuously embedded into ∈ 1 C[a,b], that is, for some M>0 and any f W2 (a,b), | |≤ ≤ + max f(x) M f W 1(a,b) M f L2(a,b) f L2(a,b) . (5.8) a≤t≤b 2

∈ (n) ∈ 1 = Since f C{mn}(A) (C(mn)(A)). Then, obviously, f W2 (a,b) for any n 0,1,2,.... Therefore, for an arbitrary n = 0,1,2,..., (n) ≤ ≤ (n) + (n+1) max f (x) M f W 1(a,b) M f L2(a,b) f L2(a,b) a≤t≤b 2 ≤ (n) + (n+1) M f L2(−∞,∞) f L2(−∞,∞) n n+1 ≤ M cα mn +cα mn+1 (by (DI)) (5.9) n n+1 n n n ≤ M cα mn +cα Lγ mn = Mc 1+Lαγ α mn

(considering that γ>1, there is a c1 > 0 such that γ>1, c1 > 0) n ≤ c1(γα) mn,n= 0,1,2,.... Based on this Lemma, we obtain the following proposition. Proposition 5.3. { }∞ Let mn n=0 be a sequence of positive numbers satisfying (WGR) and (DI).Iff ∈ L2(−∞,∞) is such that, for some (any) 0

Corollary 5.4. Let 0 <β<∞.Iff ∈ L2(−∞,∞) is such that, for some (any) 0

∞ 2 | |1/β f(x)ˆ e2t x dx < ∞, (5.12) −∞ there is a representative f(·) of the equivalence class f such that f(·) ∈ C∞(−∞,∞), ∞ 2 f (n)(x) dx < ∞,n= 0,1,2,..., ∞ (5.13) f(·) ∈ Ᏹ{β}(−∞,∞) Ᏹ(β)(−∞,∞) .

In particular, for β = 1, we obtain suﬃcient conditions for the real analyticity and entireness. 3234 MARAT V. MARKIN

6. Remarks. It is to be noted that, in [10] (see also [8, 9]), not only were equalities (2.6) for a normal operator in a complex Hilbert space proved to hold in the set-theoretical sense but also in the topological sense, the sets C{mn}(A) and C(mn)(A) considered as the inductive and, respectively, projective limits of the Banach spaces = ∈ ∞ |∃ n ≤ n = Cα[mn](A) : f C (A) c>0: A f cα mn,n 0,1,... , (6.1)

0 <α<∞, with the norms Anf f := sup (6.2) Cα[mn](A) n n≥0 α mn | | | | and the sets t>0 D(T(t A )) and t>0 D(T(t A )) as the inductive and, respectively, projective limits of the Hilbert spaces Ht[T](A) := D T(t|A|) , 0

Observe also that, in [11](seealso[10]), similar results were obtained for the generator of a bounded analytic semigroup in a Banach space.

Acknowledgments. The author would like to express his sincere affection and utmost appreciation to his teachers and superb mathematicians: the Associate of the National Academy of Sciences of Ukraine, Professor Miroslav L. Gorbachuk, D.Sc., Head of the Department of Partial Differential Equations, Institute of Mathematics, National Academy of Sciences of Ukraine (Kiev, Ukraine) and his wife and collaborator of many years, Valentina I. Gorbachuk, D.Sc., Leading Scientific Researcher, Department of Par- tial Differential Equations, Institute of Mathematics, National Academy of Sciences of Ukraine, to whom this paper is humbly dedicated.

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[8] V.I.Gorbachuk,Spaces of infinitely differentiable vectors of a nonnegative selfadjoint operator, Ukrainian Math. J. 35 (1983), 531–535. [9] V.I.GorbachukandM.L.Gorbachuk,Boundary Value Problems for Operator Differential Equations, Mathematics and its Applications (Soviet Series), vol. 48, Kluwer Aca- demic Publishers Group, Dordrecht, 1991. [10] V. I. Gorbachuk and A. V. Knyazyuk, Boundary values of solutions of operator-differential equations, Russian Math. Surveys 44 (1989), 67–111. [11] A. V. Knyazyuk, Boundary values of infinitely differentiable semigroups, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint (1985), no. 69, 47 (Russian). [12] H. Komatsu, Ultradistributions. I. Structure theorems and a characterization,J.Fac.Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105. [13] , Ultradistributions and hyperfunctions, Hyperfunctions and Pseudo-Differential Equations (Proc. Conf. on the Theory of Hyperfunctions and Analytic Functionals and Applications, RIMS, Kyoto University, Kyoto, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., vol. 287, Springer, Berlin, 1973, pp. 164– 179. [14] , Microlocal analysis in Gevrey classes and in complex domains, Microlocal Analy- sis and Applications (Montecatini Terme, 1989), Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 161–236. [15] S. Mandelbrojt, Series de Fourier et Classes Quasi-Analytiques de Fonctions, Gauthier-Villars, Paris, 1935 (French). [16] M. V. Markin, On an abstract evolution equation with a spectral operator of scalar type, Int. J. Math. Math. Sci. 32 (2002), no. 9, 555–563. [17] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. [18] R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Math- ematical Society Colloquium Publications, vol. 19, American Mathematical Society, New York, 1934. [19] A. I. Plesner, Spectral Theory of Linear Operators, Izdat. “Nauka”, Moscow, 1965. [20] S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Translations of Mathematical Monographs, vol. 7, American Mathematical Society, Rhode Island, 1963. [21] J. Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355– 361. [22] K. Yosida, Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften, vol. 123, Academic Press, New York, 1965. [23] R. J. Zimmer, Essential Results of Functional Analysis, Chicago Lectures in Mathematics, University of Chicago Press, Illinois, 1990.

Marat V. Markin: Department of Partial Diﬀerential Equations, Institute of Mathematics, Na- tional Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kiev, Ukraine 01601 E-mail address: [email protected] Mathematical Problems in Engineering

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