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 PaleyWiener type is considered as an application.
2000 Mathematics Subject Classification: 47B40, 47B15, 47B25, 30D60.
1. Introduction. As was shown in [8] (see also [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 C{mn } (A) = f ∈ C ∞ (A) | ∃α > 0, ∃c > 0 : An f ≤ cαn mn , n = 0, 1, 2, . . . ,
def C(mn ) (A) = f ∈ C ∞ (A) | ∀α > 0 ∃c > 0 : An f ≤ cαn mn , n = 0, 1, 2, . . .
(2.2)
are called the Carleman classes of vectors of the operator A corresponding to the sequence {mn }∞
n=0 of Roumie’s and Beurling’s types, respectively.
Obviously, the inclusion
C(mn ) (A) ⊆ C{mn } (A)
(2.3)
holds.
For mn := [n!]β (or, due to Stirling’s formula, for mn := nβn ), 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
Cαn ≤ mn ,
n = 0, 1, 2, . . . .
(2.4)
Note that the name WGR originates from the words “weak growth.”
Under this condition, the numerical function
T (λ) := m0
∞
λn
,
mn
n=0
0 ≤ λ < ∞, 00 := 1 ,
(2.5)
first introduced by Mandelbrojt [15], is well defined.
This function is nonnegative, continuous, and increasing.
As established in [8] (see also [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) =
t>0
C(mn ) (A) =
D T (t|A|) ,
D T (t|A|) ,
(2.6)
t>0
the normal operators T (t|A|) (0 < t < ∞) being defined in the sense of the spectral
operational calculus for a normal operator:
T (t|λ|) dEA ,
σ (A)
T 2 (t|λ|) dEA (λ)f , f < ∞ ,
D T (t|A|) := f ∈ H |
T (t|A|) :=
(2.7)
σ (A)
where the function T (·) can be replaced by any nonnegative, continuous, and increasing
function L(·) defined on [0, ∞) such that
c1 L γ1 λ ≤ T (λ) ≤ c2 L γ2 λ ,
λ > R,
(2.8)
with some positive γ1 , γ2 , c1 , c2 , and a nonnegative R.
In particular, T (·) in (2.6) is replaceable by
S(λ) := m0 sup
n≥0
λn
,
mn
0 ≤ λ < ∞,
(2.9)
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
3221
or
P (λ) := m0
∞
λ2n
2
mn
n=0
1/2
,
0 ≤ λ < ∞,
(2.10)
(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)



(x) ≤ cαn mn , n = 0, 1, 2, . . . ,
def max a≤x≤b f
C{mn } (I) =  f (·) ∈ C ∞ (I) | ∀[a, b] ⊆ I, ∀α > 0, ∃c > 0 :






maxa≤x≤b f (n) (x) ≤ cαn mn , n = 0, 1, 2, . . .
(2.11)
are the Carleman classes of ultradifferentiable functions of Roumie’s and Beurling’s
types, respectively, [1, 12, 13, 14].
In particular, for mn := [n!]β (or, due to Stirling’s formula, for mn := nβn ), 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, as is
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) =
C
F (λ) dEA (λ) =
σ (A)
F (λ) dEA (λ)
(2.13)
3222
MARAT V. MARKIN
is defined as follows:
F (A)f := lim Fn (A)f , f ∈ D F (A) ,
n→∞
D F (A) := f ∈ X | lim Fn (A)f exists
(2.14)
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) :=
σ (A)
Fn (λ) dEA (λ),
n = 1, 2, . . . ,
(2.16)
being the integrals of bounded Borel measurable functions on σ (A), are bounded 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 ∗ ) is bounded. Indeed, δ being an arbitrary Borel subset of
σ (A), [3],
v f , g ∗ , σ (A)
≤ 4 sup EA (δ)f , g ∗ ≤ 4 sup EA (δ)f g ∗ (by (2.17))
δ⊆σ (A)
≤ 4Mf g ∗ .
δ⊆σ (A)
(2.18)
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
3223
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
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 ∗ ,
σ (A)
|F (λ)| dv f , g ∗ , λ < ∞,
(2.19)
(ii)
sup
{g ∗ ∈X ∗ |g ∗ =1} {λ∈σ (A)||F (λ)|>n}
|F (λ)| dv f , g ∗ , λ → 0 as n → ∞.
(2.20)
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
∗ ≤ 4 sup F (λ) d EA (λ)f , g δ⊆σ
δ
∗ χ
(λ)F
(λ)
d
E
(λ)f
,
g
= 4 sup δ
A
σ
δ⊆σ
χδ (λ)F (λ) dEA (λ)f , g ∗ = 4 sup σ
δ⊆σ
= 4 sup EA (δ)EA (σ )F (A)f , g ∗ (by the properties of the o.c.)
(by the properties of the o.c.)
δ⊆σ
≤ 4 sup EA (δ)EA (σ )F (A)f g ∗ δ⊆σ
≤ 4 sup 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 EA σ (A) F (A)f g ∗ (since EA (σ (A)) = I (I is the identity operator in X))
∗
≤ 4M F (A)f g .
(2.22)
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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 < t < ∞) defined in the sense of
the operational calculus for a scalar type spectral operator and the function T (·) being
replaceable by any nonnegative, continuous, and increasing function L(·) defined on
[0, ∞) such that
c1 L γ1 λ ≤ T (λ) ≤ c2 L γ2 λ ,
λ > 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) by a nonnegative, continuous, and increasing function satisfying (3.1) with some positive γ1 , γ2 , c1 , c2 , and a
nonnegative R ≥ 0.
Let
f∈
t>0
T (t|A|)
T (t|A|) .
(3.2)
t>0
Then, for some (any) 0 < t < ∞, f ∈ D(T (t|A|)), which, according to Proposition 2.1,
implies, in particular, that, for any g ∗ ∈ X ∗ ,
T (t|λ|) dv f , g ∗ , λ < ∞.
(3.3)
L γ1 t|λ| dv f , g ∗ , λ < ∞.
(3.4)
σ (A)
For any g ∗ ∈ X ∗ ,
σ (A)
Indeed,
L γ1 t|λ| dv f , g ∗ , λ
=
L γ1 t|λ| dv f , g ∗ , λ +
L γ1 t|λ| dv f , g ∗ , λ
{λ∈σ (A)|t|λ|≤R}
{λ∈σ (A)|t|λ|>R}
≤ L γ1 R v f , g ∗ , σ (A) +
L γ1 t|λ| dv f , g ∗ , λ (by (2.18))
{λ∈σ (A)|t|λ|>R}
≤ L γ1 R 4Mf g ∗ +
L γ1 t|λ| dv f , g ∗ , λ (by (3.1))
{λ∈σ (A)|t|λ|>R}
1
F (t|λ|) dv f , g ∗ , λ
≤ L γ1 R 4Mf g ∗ +
c1 {λ∈σ (A)|t|λ|>R}
1
F (t|λ|) dv f , g ∗ , λ (by (3.3))
≤ L γ1 R 4Mf g ∗ +
c1 σ (A)
σ (A)
< ∞.
(3.5)
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
3225
Further,
sup
{g ∗ ∈X ∗ |g ∗ =1} {λ∈σ (A)|t|λ|≤R, L(γ1 t|λ|)>n}
L γ1 t|λ| dv f , g ∗ , λ = 0
(3.6)
for all sufficiently large natural n’s since, when t|λ| ≤ R, L(γ1 t|λ|) ≤ L(γ1 R).
On the other hand,
{λ∈σ (A)|t|λ|>R, L(γ1 t|λ|)>n}
L γ1 t|λ| dv f , g ∗ , λ
(by (3.1))
1
T (t|λ|) dv f , g ∗ , λ (by (2.21))
c1 {λ∈σ (A)|t|λ|>R, T (t|λ|)>c1 n}
1
EA λ ∈ σ (A) | T (t|λ|) > c1 n T (t|A|)f g ∗ ≤
c1
(by the continuity of the s.m.)
≤
(3.7)
→ 0 as n → ∞.
Therefore, by Proposition 2.1, f ∈ D(L(γ1 t|A|)).
Thus, we have proved the inclusions
D T (t|A|) ⊆
D L(t|A|) ,
t>0
D T (t|A|) ⊆
D L(t|A|) .
t>0
t>0
(3.8)
t>0
Similarly, one can derive from (3.1) the inverse inclusions:
D T (t|A|) ⊇
D L(t|A|) ,
t>0
D T (t|A|) ⊇
D L(t|A|) .
t>0
t>0
(3.9)
t>0
Thus,
D T (t|A|) =
D L(t|A|) ,
t>0
D T (t|A|) =
D L(t|A|) .
t>0
t>0
t>0
(3.10)
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 A f ≤ cαn mn ,
n = 0, 1, 2, . . . .
(3.11)
For any g ∗ ∈ X ∗ ,
∞
1
|λ|n
∗
|λ| dv f , g , λ =
T
dv f , g ∗ , λ
n
n
2α
σ (A)
σ (A) n=0 2 α mn
(by the monotone convergence theorem)
=
=
≤
∞ |λ|n
2n αn mn
n=0 σ (A)
∞
1
n αn m
2
n
n=0
σ (A)
dv f , g ∗ , λ
|λ|n dv f , g ∗ , λ
(by (2.22))
∞
1
4M An f g ∗ (by (3.11))
n αn m
2
n
n=0
≤ 4Mc
∞
1 g ∗ = 8Mc g ∗ < ∞.
n
2
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
∞
nk
T 1 |A| EA ∆n ≤ 4M
k
2α
2 αk mk
k=0
by condition (WGR), there is a C = C(α, n) > 0 :
≤ 4MC
∞
1
= 8MC.
k
2
k=0
nk
≤ C, k = 0, 1, . . .
k
α mk
(3.14)
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
3227
For any 1 ≤ m < n,
T 1 |A| EA ∆n f − T 1 |A| EA ∆m f , g ∗ 2α
2α
(by the properties of the o.c.)
1
T
|λ| dEA (λ)f , g ∗ 2α
{λ∈σ (A)|m<|λ|≤n}
(by the properties of the o.c.)
1
∗ T
(λ)f
,
g
=
|λ|
d
E
A
2α
{λ∈σ (A)|m<|λ|≤n}
1
|λ| dv f , g ∗ , λ (by (3.12))
T
≤
2α
{λ∈σ (A)|m<|λ|}
→ 0
(3.15)
as m → ∞.
Since a reflexive Banach space is weakly complete (see, e.g., [3]), we infer that the sequence {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
|A| .
f ∈D T
2α
(3.17)
Therefore,
f∈
D T (t|A|)
t>0
D T (t|A|) , resp. ,
(3.18)
t>0
which proves the inclusions
C{mn } (A) ⊆
t>0
C(mn ) (A) ⊆
t>0
Now, we are to prove the inverse inclusions.
D T (t|A|) ,
D T (t|A|) .
(3.19)
3228
MARAT V. MARKIN
Let
f∈
D T (t|A|)
t>0
D T (t|A|) .
(3.20)
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 ∗ ,
σ (A)
∞
[t|λ|]k
dv f , g ∗ , λ
mk
σ (A)
k=0
T (t|λ|) dv f , g ∗ , λ
=
tN
|λ|N dv f , g ∗ , λ ≤
mN
(3.21)
σ (A)
(by Proposition 2.1),
< ∞.
Further, for any N = 0, 1, 2, . . . ,
sup
{g ∗ ∈X ∗ |g ∗ =1} {λ∈σ (A)|(t N /mN )|λ|N >n}
≤
sup
tN
|λ|N dv f , g ∗ , λ
mN
{g ∗ ∈X ∗ |g ∗ =1} {λ∈σ (A)|T (t|λ|)>n}
→ 0
T (t|λ|) dv f , g ∗ , λ
(by Proposition 2.1),
as n → ∞.
(3.22)
By Proposition 2.1, (3.21) and (3.22) imply that
f ∈ C ∞ (A).
(3.23)
Further, by (2.22),
sup
{g ∗ ∈X ∗ |g ∗ =1} σ (A)
T (t|λ|) dv f , g ∗ , λ
(by (2.22))
≤ 4M T (t|A|)f < ∞.
(3.24)
By (2.22),
0<c:=
sup
{g ∗ ∈X ∗ |g ∗ =1} σ (A)
T (t|λ|) dv f , g ∗ , λ + 1
≤ 4M T (t|A|)f < ∞.
(3.25)
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
3229
Whence, for any n = 0, 1, 2, . . . ,
tn
|λ|n dv f , g ∗ , λ
c≥
sup
{g ∗ ∈X ∗ |g ∗ =1} σ (A) mn
tn
n
∗ ≥
sup
λ
d
E
(λ)f
,
g
A
mn {g∗ ∈X ∗ |g∗ =1} σ (A)
≥
n
t
mn
sup
{g ∗ ∈X ∗ |g ∗ =1}
σ (A)
λn dEA (λ)f , g ∗ (by the properties of the o.c.)
(3.26)
(by the properties of the o.c.)
=
tn
mn
sup
{g ∗ ∈X ∗ |g ∗ =1}
n
A f , g∗ (as follows from the Hahn-Banach theorem)
tn A n f .
=
mn
Thus, for some (any) t > 0,
n
n A f ≤ c 1 mn ,
t
n = 0, 1, 2, . . . .
(3.27)
Hence,
f ∈ C{mn } (A)
C(mn ) (A), resp. ,
(3.28)
which proves the inverse inclusions
C{mn } (A) ⊇
t>0
C(mn ) (A) ⊇
D T (t|A|) ,
D T (t|A|) .
(3.29)
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
D T (t|A|) ,
C{mn } (A) ⊆
t>0
C(mn ) (A) ⊆
D T (t|A|)
(3.30)
t>0
only.
The inverse inclusions
C{mn } (A) ⊇
t>0
C(mn ) (A) ⊇
D T (t|A|) ,
D T (t|A|)
t>0
hold regardless whether X is reflexive or not.
(3.31)
3230
MARAT V. MARKIN
4. The Gevrey classes of a scalar type spectral operator. Let 0 < β < ∞. As is easily
seen, the sequence mn = [n!]β , n = 0, 1, 2, . . . , satisfies condition (WGR) and, thus, the
function
T (λ) :=
∞
λn
,
β
[n!]
n=0
0 ≤ λ < ∞,
(4.1)
is well defined.
According to Stirling’s formula,
nβn ∼ (2π n)−β/2 eβn [n!]β
as n → ∞.
(4.2)
Hence, there is such a C = C(β) ≥ 1 such that
[n!]β ≤ nβn ≤ C(2π n)−β/2 eβn [n!]β ≤ Ceβn [n!]β ,
n = 0, 1, 2, . . . .
(4.3)
n
∞
∞ β n
∞
e λ
λn
λn
1 2eβ λ
≤
≤
T
(λ)
≤
C
=
C
nβn n=0 nβn
nβn
2n nβn
n=0
n=0
β n ∞
β n
1
2e λ
2e λ
≤ C sup
= 2C sup
, 0 ≤ λ < ∞.
βn
n
n
2
nβn
n≥0
n≥0
n=0
(4.4)
Taking this into account, we infer
sup
n≥0
Now, we consider the family of functions
ρλ (x) :=
λx
,
x βx
0 ≤ x < ∞, 1 ≤ λ < ∞ 00 := 1 .
(4.5)
It is easy to make sure that the function ρλ (·) attains its maximum value on [0, ∞) at
the point xλ = e−1 λ1/β .
Therefore,
sup
n≥0
λn
λx
−1 1/β
≤ sup βx = ρλ xλ = eβe λ .
nβn
x
x≥0
(4.6)
For λ ≥ eβ , let N be the integer part of xλ = e−1 λ1/β .
Hence, N ≥ 1 and
sup
n≥0
λN
λn
≥
= exp N ln λ − βN ln N
nβn
N βN
1
−1 1/β
≥ exp xλ − 1 ln λ − βxλ ln xλ = eβe λ ,
λ
(4.7)
β
λ≥e .
Obviously, for all sufficiently large positive λ’s,
e−(βe
−1 /2)λ1/β
≤
1
.
λ
(4.8)
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,
β (e−β /2β )λ)1/β
e(β
≤ T (λ) ≤ 2C sup
n≥0
= 2Ce
β n
2e λ
≤ 2C sup ρ2eβ λ (x)
nβn
x≥0
1/β
βe−1 (2eβ λ)
≤e
(4ββ λ)1/β
(4.9)
.
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 Banach space and 0 < β < ∞. Then
Ᏹ{β} (A) =
Ᏹ
(β)
(A) =
1/β
D et|A|
,
t>0
1/β
D et|A|
.
(4.10)
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] (see also [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
Ᏹ
(β)
(A) ⊇
1/β
D et|A|
.
(4.11)
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
f ∈ C{mn } (A) C(mn ) (A) ⇐⇒
∞
−∞
fˆ(x)2 T 2 (t|x|) dx < ∞
(5.1)
(fˆ is the Fourier transform of f ) for some (any) 0 < t < ∞, the function T (·) being
replaceable by any nonnegative, continuous, and increasing function L(·) defined on
[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 relative to the differential operator A in L2 (−∞, ∞) reveals itself as the smoothness in the
ordinary sense.
3232
MARAT V. MARKIN
For any f ∈ W2n (I), where I is an interval of the real axis and W2n (I) = H n (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
f ∈ W2∞ (−∞, ∞) :=
∞
W2n (−∞, ∞),
(5.2)
n=0
let f (·) be the infinite-differentiable representative of the equivalence class f such that
∞
−∞
(n) 2
f (t) dt < ∞,
n = 0, 1, 2, . . . .
(5.3)
Let
def
Ĉ{mn } (−∞, ∞) =
def
Ĉ(mn ) (−∞, ∞) =
f ∈ W2∞ (−∞, ∞) | ∀[a, b] ⊆ (−∞, ∞) ∃α > 0,
∃c > 0 : max f (n) (t) ≤ cαn mn , n = 0, 1, 2, . . . ,
a≤t≤b
f∈
(5.4)
W2∞ (−∞, ∞)
| ∀[a, b] ⊆ (−∞, ∞), ∀α > 0
(n) ∃c > 0 : max f (t) ≤ cαn 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 > 0 and a γ > 1 such that
mn+1 ≤ Lγ n 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 (·).
Observe that, for 0 ≤ β < ∞, the Gevrey sequence mn = [n!]β , n = 0, 1, 2, . . . , meets
condition (DI) with any γ > 1. Indeed, in this case, mn+1 /mn = (n + 1)β , n = 0, 1, 2, . . . .
Lemma 5.2. Let a sequence of positive numbers {mn }∞
n=0 satisfy condition (DI). Then
C{mn } (A) ⊆ Ĉ{mn } (−∞, ∞),
(5.5)
C(mn ) (A) ⊆ Ĉ(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
f L2 (−∞,∞) =
∞
−∞
(n)
f (x)2 dx
1/2
≤ cαn mn ,
n = 0, 1, 2, . . . .
(5.7)
3233
THE CARLEMAN CLASSES OF A SCALAR OPERATOR
We fix a finite segment [a, b] of the real axis. Then, according to the Sobolev embedding
theorems [20] (see also [22, 23]), the space W21 (a, b) is continuously embedded into
C[a, b], that is, for some M > 0 and any f ∈ W21 (a, b),
!
"
max |f (x)| ≤ Mf W 1 (a,b) ≤ M f L2 (a,b) + f L2 (a,b) .
(5.8)
2
a≤t≤b
Since f ∈ C{mn } (A) (C(mn ) (A)). Then, obviously, f (n) ∈ W21 (a, b) for any n = 0, 1, 2, . . . .
Therefore, for an arbitrary n = 0, 1, 2, . . . ,
!
"
max f (n) (x) ≤ Mf W 1 (a,b) ≤ M f (n) L2 (a,b) + f (n+1) L2 (a,b)
2
a≤t≤b
!
"
≤ M f (n) L2 (−∞,∞) + f (n+1) L2 (−∞,∞)
#
$
≤ M cαn mn + cαn+1 mn+1
(by (DI))
# n
$
#
$
n+1
n
Lγ mn = Mc 1 + Lαγ n αn mn
≤ M cα mn + cα
(5.9)
(considering that γ > 1, there is a c1 > 0 such that γ > 1, c1 > 0)
≤ c1 (γα)n 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)
2
and (DI). If f ∈ L (−∞, ∞) is such that, for some (any) 0 < t < ∞,
∞
−∞
fˆ(x)2 T 2 (t|x|) dx < ∞,
(5.10)
there is a representative f (·) of the equivalence class f such that f (·) ∈ C ∞ (−∞, ∞),
∞
(n)
f (x)2 dx < ∞, n = 0, 1, 2, . . . ,
f (·) ∈ C{mn } (−∞, ∞) C(mn ) (−∞, ∞) ,
−∞
(5.11)
the function T (·) being replaceable by any nonnegative, continuous, and increasing function L(·) defined on [0, ∞) and satisfying (3.1) with some positive γ1 , γ2 , c1 , c2 , and a
nonnegative R.
Corollary 5.4. Let 0 < β < ∞. If f ∈ L2 (−∞, ∞) is such that, for some (any) 0 < t < ∞,
∞
−∞
fˆ(x)2 e2t|x|1/β dx < ∞,
(5.12)
there is a representative f (·) of the equivalence class f such that f (·) ∈ C ∞ (−∞, ∞),
∞
(n)
f (x)2 dx < ∞,
n = 0, 1, 2, . . . ,
f (·) ∈ Ᏹ{β} (−∞, ∞) Ᏹ(β) (−∞, ∞) .
∞
(5.13)
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
Cα[mn ] (A) := f ∈ C ∞ (A)| ∃c > 0 : An f ≤ cαn mn , n = 0, 1, . . . ,
(6.1)
0 < α < ∞, with the norms
f Cα[mn ] (A) := sup
n≥0
n A f (6.2)
αn 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 < t < ∞,
(6.3)
with inner products
(f , g)Ht[T ] (A) := T (t|A|)f , T (t|A|)g ,
0 < t < ∞.
(6.4)
Observe also that, in [11] (see also [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 Partial Differential Equations, Institute of Mathematics, National Academy of Sciences of
Ukraine, to whom this paper is humbly dedicated.
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Marat V. Markin: Department of Partial Differential Equations, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kiev, Ukraine 01601
E-mail address: mmarkin@comcast.net