On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups
Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C₀-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator.
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Інститут математики НАН України
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irk-123456789-1647502020-02-11T01:26:41Z On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups Markin, M.V. Статті Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C₀-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator. Знайдено необхідні та достатні умови для того, щоб спектральний оператор скалярного типу в банаховому просторі породжував ультрадиференційовну C₀-напівгрупу Карлемана. Ці умови сформульовано виключно у термінах спектра оператора. 2008 Article On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups / M.V. Markin // Український математичний журнал. — 2008. — Т. 60, № 9. — С. 1215–1233. — Бібліогр.: 31 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164750 531.19 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Markin, M.V. On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups Український математичний журнал |
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Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C₀-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator. |
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Article |
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Markin, M.V. |
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Markin, M.V. |
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Markin, M.V. |
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On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups |
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On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups |
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On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups |
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On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups |
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On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups |
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on scalar-type spectral operators and carleman ultradifferentiable c₀-semigroups |
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Інститут математики НАН України |
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2008 |
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| citation_txt |
On scalar-type spectral operators and Carleman ultradifferentiable C₀-semigroups / M.V. Markin // Український математичний журнал. — 2008. — Т. 60, № 9. — С. 1215–1233. — Бібліогр.: 31 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT markinmv onscalartypespectraloperatorsandcarlemanultradifferentiablec0semigroups |
| first_indexed |
2025-07-14T17:20:31Z |
| last_indexed |
2025-07-14T17:20:31Z |
| _version_ |
1837643733772795904 |
| fulltext |
UDC 531.19
M. V. Markin (Fresno, CA, USA)
ON SCALAR-TYPE SPECTRAL OPERATORS
AND CARLEMAN ULTRADIFFERENTIABLE C0-SEMIGROUPS
ПРО СПЕКТРАЛЬНI ОПЕРАТОРИ СКАЛЯРНОГО ТИПУ
ТА УЛЬТРАДИФЕРЕНЦIЙОВНI C0-НАПIВГРУПИ
КАРЛЕМАНА
Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator
of a Carleman ultradifferentiable C0-semigroup are found. The conditions are formulated exclusively in
terms of the operator’s spectrum.
Знайдено необхiднi та достатнi умови для того, щоб спектральний оператор скалярного типу в
банаховому просторi породжував ультрадиференцiйовну C0-напiвгрупу Карлемана. Цi умови сфор-
мульовано виключно у термiнах спектра оператора.
1. Introduction. This paper is a natural sequal to [1, 2], where criteria of a scalar-
type spectral operator in a complex Banach space being a generator of a C0-semigroup,
an analytic C0-semigroup, an infinite differentiable, or a Gevrey ultradifferentiable C0-
semigroup were found.
Here, we are to generalize the results of [2] concerning the Gevrey ultradifferentia-
bility by obtaining necessary and sufficient conditions for a scalar-type spectral operator
in a complex Banach space to be a generator of a Carleman ultradifferentiable C0-
semigroup.
It is to be noted that such conditions, as well as those of [1, 2], will be formulated
exclusively in terms of the operator’s spectrum, no restrictions on its resolvent behavior
necessary. This fact appears to be distinctive for scalar-type spectral operators making
the results significantly more transparant than in general case [3 – 7] (see also [8, 9]) and
purely qualitative.
Similar results for a normal operator in a complex Hilbert space are discussed in a
more general context in [10 – 13].
2. Preliminaries. 2.1. Scalar-type spectral operators. Henceforth, unless specified
otherwise, A is a scalar-type spectral operator in a complex Banach space X with a
norm ‖·‖ and EA(·) is its spectral measure (the resolution of the identity), the operator’s
spectrum σ(A) being the support for the latter [14, 15].
Note that, in a Hilbert space, the scalar-type spectral operators are those similar to
the normal ones [16].
For such operators, there has been developed an operational calculus for Borel
measurable functions on C (σ(A)) [14, 15], F (·) being such a function, a new scalar-
type spectral operator
F (A) =
∫
C
F (λ) dEA(λ) =
∫
σ(A)
F (λ) dEA(λ) (2.1)
is defined as follows:
c© M. V. MARKIN, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9 1215
1216 M. V. MARKIN
F (A)f := lim
n→∞
Fn(A)f, f ∈ D(F (A)),
D(F (A)) :=
{
f ∈ X
∣∣ lim
n→∞
Fn(A)f exists
}
(D(·) is the domain of an operator), where
Fn(·) := F (·)χ{λ∈σ(A) | |F (λ)|≤n}(·), n = 1, 2, . . .
(χα(·) is the characteristic function of a set α), and
Fn(A) :=
∫
σ(A)
Fn(λ) dEA(λ), n = 1, 2, . . . ,
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., [17, 18]).
The properties of the spectral measure, EA(·), and the operational calculus under-
lying the entire subsequent discourse are exhaustively delineated in [14, 15]. Let’s just
outline here a few facts that will be especially important for us.
Observe first that, due to its strong countable additivity, the spectral measure EA(·)
is bounded, i.e., there is an M > 0 such that, for any Borel set δ in C [19],
‖EA(δ)‖ ≤ M. (2.2)
Note that, in (2.2), the notation ‖ · ‖ was used to designate the norm in the space of
bounded linear operators on X. We shall 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.
As we saw [2, 20], 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 is the
pairing between the space X and its dual, X∗) is bounded. Indeed,
v(f, g∗, σ(A)) ≤ 4M‖f‖‖g∗‖. (2.3)
Also [2, 20], F (·) being an arbitrary Borel measurable function on C (σ(A)), for any
f ∈ D(F (A)), g∗ ∈ X∗ and arbitrary Borel sets δ ⊆ σ,∫
σ
|F (λ)| dv(f, g∗, λ) ≤ 4M‖EA(σ)‖‖F (A)f‖‖g∗‖. (2.4)
In particular, ∫
σ(A)
|F (λ)| dv(f, g∗, λ) ≤ 4M‖F (A)f‖‖g∗‖. (2.5)
Observe also that, as follows from [1, 21, 22], if a scalar-type spectral operator A gen-
erates a C0-semigroup, it’s of the form
{
etA
∣∣ t ≥ 0
}
, where the operator exponentials
are defined in accordance with the operational calculus (2.1).
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.
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ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1217
2.2. Carleman ultradifferentiability. Let X be a Banach space with a norm ‖ · ‖,
I be an interval of the real axis, C∞(I, X) be the set of all X-valued functions strongly
infinite differentiable on I, and
{
mn
}∞
n=0
be a sequence of positive numbers.
The sets
C{mn}(I,X) df=
{
g(·) ∈ C∞(I,X)
∣∣ ∀[a, b] ⊆ I ∃α > 0 ∃c > 0:
max
a≤t≤b
‖g(n)(t)‖ ≤ cαnmn, n = 0, 1, 2, . . .
}
and
C(mn)(I,X) df=
{
g(·) ∈ C∞(I,X)
∣∣ ∀[a, b] ⊆ I ∀α > 0 ∃c > 0:
max
a≤t≤b
‖g(n)(t)‖ ≤ cαnmn, n = 0, 1, 2, . . .
}
are called the Carleman classes of strongly ultradifferentiable functions corresponding
to the sequence
{
mn
}∞
n=0
of Roumieu’s and Beurling’s types, respectively (for numeric
functions, see [23 – 25]).
Obviously,
C(mn)(I, X) ⊆ C{mn}(I,X).
Observe that, for mn := [n!]β or, due to Stirling’s formula, mn := nβn, n = 0, 1, 2, . . . ,
0 ≤ β < ∞, we obtain the well-known Gevrey classes, E{β}(I, X) and E(β)(I, X) (for
numeric functions, see [26]). In particular, E{1}(I,X) and E(1)(I,X) are the classes of
real analytic and entire vector functions, respectively.
2.3. Carleman classes of vectors. Let
C∞(A) df=
∞⋂
n=0
D(An).
The vector sets
C{mn}(A) df=
{
f ∈ C∞(A)
∣∣ ∃α > 0 ∃c > 0:
‖Anf‖ ≤ cαnmn, n = 0, 1, 2, . . .
}
and
C(mn)(A) df=
{
f ∈ C∞(A)
∣∣ ∀α > 0 ∃c > 0:
‖Anf‖ ≤ cαnmn, n = 0, 1, 2, . . .
}
are called the Carleman classes of the operator A corresponding to the sequence
{
mn
}∞
n=0
of Roumie’s and Beurling’s types, respectively. Again
C(mn)(A) ⊆ C{mn}(A). (2.6)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1218 M. V. MARKIN
For mn := [n!]β or mn := nβn, n = 0, 1, 2, . . . , 0 ≤ β < ∞, the above are the
Gevrey classes of the operator A, E{β}(A) and E(β)(A) (see, e.g., [27 – 29]). In par-
ticular, E{1}(A) and E(1)(A) are the celebrated classes of analytic and entire vectors,
respectively [30, 31].
3. The sequence
{
mn
}∞
n=0
. The sequence
{
mn
}∞
n=0
being subject to the con-
dition
(WGR) for any α > 0, there exist such a c = c(α) > 0 that
cαn ≤ mn, n = 0, 1, 2, . . . ,
the scalar function
T (λ) := m0
∞∑
n=0
λn
mn
, 0 ≤ λ < ∞, 00 := 1, (3.1)
first introduced by S. Mandelbrojt [25] is well-defined (see also [29]).
The function T (·) is, evidently, continuous, strictly increasing and T (0) = 1. When-
ever the function T (·) is well defined, so is
M(λ) := lnT (λ), 0 ≤ λ < ∞. (3.2)
The latter is also continuous, strictly increasing and M(0) = 0. Thus, it has an inverse
M−1(·) defined on [0,∞) and inheriting all the aforementioned properties of M(·).
According to [20], for a scalar-type spectral operator A in a complex Banach space
X and 0 < β < ∞, we have
C{mn}(A) ⊇
⋃
t>0
D(T (t|A|)),
C(mn)(A) ⊇
⋂
t>0
D(T (t|A|)),
(3.3)
the function T (·) being replaceable by any nonnegative, continuous, and increasing
function L(·) defined on [0,∞) such that
c1L(γ1λ) ≤ T (λ) ≤ c2L(γ2λ), λ > R,
with some positive γ1, γ2, c1, c2, and a nonnegative R.
In particular, T (·) in (3.3) is replaceable by [29]
S(λ) := m0 sup
n≥0
λn
mn
, 0 ≤ λ < ∞,
or
P (λ) := m0
[ ∞∑
n=0
λ2n
m2
n
]1/2
, 0 ≤ λ < ∞.
Observe that inclusions (3.3) turn into equalities provided the space X is reflex-
ive [20].
The positive sequence
{
mn
}∞
n=0
will be subject to the following conditions:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1219
(GR) for some α > 0 and c > 0,
cαnn! ≤ mn, n = 0, 1, 2, . . . ;
(SBC) for some l > 0, L > 0 and h > 1, H > 1,
lhn ≤
n∑
k=0
mn
mkmn−k
≤ LHn, n = 0, 1, 2, . . . .
Obviously, condition (GR) is stronger than (WGR) and condition (SBC) resembles
the fundemental property of the binomial coefficients
n∑
k=0
(
n
k
)
= 2n, n = 0, 1, 2, . . . .
Actually, when mn = n!, n = 0, 1, 2, . . . , we positively arrive at the latter.
Observe also that there are sequences of positive numbers satisfying both (GR) and
(SBC), e.g., mn = [n!]β , n = 0, 1, 2, . . . , 1 ≤ β < ∞.
As is easily seen, the sequence mn :=
√
n!, n = 0, 1, 2, . . . , satisfies condi-
tion (SBC), but doesn’t meet condition (GR).
We leave it to the reader to make sure that the sequence
mn :=
n2n for n = n(k),
en4
otherwise,
where n(0) := 1, n(1) := 2, n(k) := n(k − 2) + n(k − 1) + 1, k = 2, 3, . . . , satisfies
condition (GR) but not (SBC).
Thus, conditions (GR) and (SBC) are independent.
Now, let’s see what conditions (GR) and (SBC) imply for the function M(·) (3.2).
By condition (GR), for a certain α > 0 and a certain c > 0,
T (λ) = m0
∞∑
n=0
λn
mn
≤ m0c
−1
∞∑
n=0
(α−1λ)n
n!
= m0c
−1eα−1λ, 0 ≤ λ < ∞.
Whence
M(λ) ≤ ln(m0c
−1) + α−1λ, 0 ≤ λ < ∞.
Therefore, there is such an R = R(α, c) > 0 that
M(λ) ≤ 2α−1λ, R ≤ λ < ∞.
Substituting M−1(λ) for λ, we arrive at the following estimate:
2α−1M−1(λ) ≥ λ, M(R) ≤ λ < ∞, (3.4)
with some α > 0 and R > 0.
Condition (SBC) implies that with some h > 1 and l > 0
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1220 M. V. MARKIN
T 2(λ) =
Cauchy’s product of series
= m2
0
∞∑
n=0
n∑
k=0
1
mkmn−k
λn ≥ m2
0l
∞∑
n=0
(hλ)n
mn
= m0lT (hλ), 0 ≤ λ < ∞.
Whence
M(λ) ≥ 2−1M(hλ) + 2−1 ln(m0l), 0 ≤ r < ∞.
Iductively, we infer that, for certain h > 1 and l > 0 and any natural n,
M(λ) ≥ 2−nM(hnλ) +
[ n∑
k=1
2−k
]
ln(m0l) =
= 2−nM(hnλ) + [1− 2−n] ln(m0l), 0 ≤ λ < ∞. (3.5)
Analogously, condition (SBC) implies that, along with (3.5) the function M(·) sat-
isfies the following estimate:
M(λ) ≤ 2−nM(Hnλ) + [1− 2−n] ln(m0L), 0 ≤ λ < ∞. (3.6)
4. Ultradifferentiability of an orbit. Let A be a scalar-type spectral operator
generating a C0-semigroup
{
etA
∣∣ t ≥ 0
}
.
Proposition 4.1. Let I be a subinterval of [0,∞) and
{
mn
}∞
n=0
be a sequence
of positive numbers. Then the restriction of an orbit etAf, 0 ≤ t < ∞, f ∈ X, to I
belongs to C{mn}(I, X) (C(mn)(I, X)) if and only if
etAf ∈ C{mn}(A) (C(mn)(A), respectively) for any t ∈ I.
Proof. “Only if” part. Assume that the restriction of an orbit etAf, 0 ≤ t < ∞,
f ∈ X, to a subinterval I of [0,∞) belongs to C∞(I,X).
Then by [2] the restriction of etAf, 0 ≤ t < ∞, f ∈ X, to I is strongly infinite
differentiable on I, i.e., etA ∈ C∞(I,X) and, for any natural n,
dn
dtn
etAf = AnetAf, t ∈ I.
Furthermore, the fact that the restriction of etAf, 0 ≤ t < ∞, f ∈ X, to I belongs
to the class C{mn}(I,X) (C(mn)(I,X)) implies that, for an arbitrary t ∈ I, a certain
(any) α > 0, and a certain c > 0:
‖AnetAf‖ = ‖ dn
dtn
etAf‖ ≤ cαnmn, n = 0, 1, . . . .
Therefore,
etAf ∈ C{mn}(A) (C(mn)(A)), t ∈ I.
“If” part. Let an orbit etAf, 0 ≤ t < ∞, f ∈ X, be such that
etAf ∈ C{mn}(A) (C(mn)(A)), t ∈ I,
where I is a subinterval of [0,∞).
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1221
Hence, for arbitrary t ∈ I and some (any) α(t) > 0, there is such a c(t, α) > 0 that
‖AnetAf‖ ≤ c(t, α)α(t)nmn, n = 0, 1, 2, . . . . (4.1)
The inclusions
C(mn)(A) ⊆ C{mn}(A) ⊆ C∞(A)
imply, by [2], that
etAf ∈ C∞(A) and
dn
dtn
etAf = AnetAf, n = 1, 2, . . . , t ∈ I. (4.2)
Let us fix an arbitrary subsegment [a, b] ⊆ I. For n = 0, 1, . . . , we have
max
a≤t≤b
∥∥∥∥ dn
dtn
etAf
∥∥∥∥ =
by (4.2);
= max
a≤t≤b
‖AnetAf‖ =
by the properties of the o.c. and the Hahn – Banach Theorem;
= max
a≤t≤b
sup
{g∗∈X∗ | ‖g∗‖=1}
∣∣∣∣∣∣∣
〈 ∫
σ(A)
λnetλ dEA(λ)f, g∗
〉∣∣∣∣∣∣∣ ≤
by the properties of the o.c.;
≤ max
a≤t≤b
sup
{g∗∈X∗ | ‖g∗‖=1}
∣∣∣∣∣∣∣
∫
σ(A)
λnetλ d〈EA(λ)f, g∗〉
∣∣∣∣∣∣∣ ≤
≤ max
a≤t≤b
sup
{g∗∈X∗ | ‖g∗‖=1}
∫
σ(A)
|λ|net Re λ dv(f, g∗, λ) =
= sup
{g∗∈X∗ | ‖g∗‖=1}
max
a≤t≤b
∫
{λ∈σ(A)|Re λ≤0}
|λ|net Re λ dv(f, g∗, λ) +
+
∫
{λ∈σ(A)|Re λ>0}
|λ|net Re λ dv(f, g∗, λ)
≤
≤ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ(A)|Re λ≤0}
|λ|nea Re λ dv(f, g∗, λ)+
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
1222 M. V. MARKIN
+ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ(A)|Re λ>0}
|λ|neb Re λ dv(f, g∗, λ) ≤
≤ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
σ(A)
|λ|nea Re λ dv(f, g∗, λ)+
+ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
σ(A)
|λ|neb Re λ dv(f, g∗, λ) ≤
by (2.5);
≤ sup
{g∗∈X∗ | ‖g∗‖=1}
4M‖AneaAf‖‖g∗‖ +
+ sup
{g∗∈X∗ | ‖g∗‖=1}
4M‖AnebAf‖‖g∗‖ =
= 4M [‖AneaAf‖+ ‖AnebAf‖] ≤
by (4.1);
≤ 4M [c(a, α) + c(b, α)]max[α(a), α(b)]nmn, n = 0, 1, 2, . . . .
This implies that the restriction of etAf, 0 ≤ t < ∞, f ∈ X, to the subinterval
I ⊆ [0, T ) belongs to the Carleman class C{mn}(I,X) (C(mn)(I,X)).
The proposition is proved.
5. Carleman ultradifferentiable C0-semigroups. Let {mn}∞n=0 be a sequence of
positive numbers. We shall call a C0-semigroup
{
S(t)
∣∣ t ≥ 0
}
in a Banach space X
a C{mn}-semigroup (a C(mn)-semigroup) if, for any f ∈ X, the orbit S(·)f belongs to
the Carleman class C{mn}((0,∞), X) (C(mn)((0,∞), X), respectively). We shall call a
C0-semigroup a Carleman ultradifferentiable semigroup if, for some positive sequence{
mn
}∞
n=1
, it is a C{mn}-semigroup or, which, due to inclusions (2.6), is the same, a
C(mn)-semigroup.
Theorem 5.1. Let a scalar-type spectral operator A generate a C0-semigroup
and a sequence of positive numbers {mn}∞n=0 satisfy conditions (GR) and (SBC). Then
the C0-semigroup is a C{mn}-semigroup if and only if there are a real a and a positive
b such that
Re λ ≤ a− bM(| Im λ|), λ ∈ σ(A).
Proof. “If” part. Consider an arbitrary orbit etAf, 0 ≤ 0 < ∞, f ∈ X.
According to Proposition 4.1, we need to show that
etAf ∈ C{mn}(A), 0 < t < ∞.
In view of inclusions (3.3), it suffies to show that
etAf ∈
⋃
s>0
D(T (s|A|)), 0 < t < ∞.
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ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1223
Let’s fix an arbitrary t > 0. Let’s also fix a sufficiently large natural N so that
2−Nγ ≤ t,
where γ := max(1, 2b−1), and set
s := H−N [2α−1 + 1]−1 > 0,
where α and H are some positive constants from estimates (3.4) and (3.6), respectively.
For any g∗ ∈ X∗, ∫
σ(A)
T (s|λ|)et Re λ dv(f, g∗, λ) =
=
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
T (s|λ|)et Re λ dv(f, g∗, λ)+
+
∫
{λ∈σ(A)|min(−M(R),a)<Re λ≤a}
T (s|λ|)et Re λ dv(f, g∗, λ) < ∞,
where R is a positive constant from (3.4).
Indeed, the latter integral is finite due to the boundedness of the set
{
λ ∈ σ(A)
∣∣
min(−M(R), a) < Re λ ≤ a
}
(note that, for a ≤ −M(R), the set is, obviously,
empty), the continuity of the integrated function, and the finiteness of the positive mea-
sure v(f, g∗, ·) (see (2.3)).
For the former of the two above integrals, we have∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
T (s|λ|)et Re λ dv(f, g∗, λ) ≤
for λ ∈ σ(A), Re λ ≤ min(−M(R), a)
Re λ ≤ −M(R) and | Im λ| ≤ M−1[b−1(a− Re λ)];
≤
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
eM(s[−Re λ+M−1(b−1(a−Re λ))])et Re λ dv(f, g∗, λ).
Let’s consider separately the two posible cases: a ≤ 0 and a > 0.
If a ≤ 0, then a− Re λ ≤ −2 Re λ for all λ’s such that Re λ ≤ min(−M(R), a),
and we have
∫
{λ∈σ(A)|Re λ≤min(−R,a)}
eM(s[−Re λ+M−1(b−1(a−Re λ))])et Re λ dv(f, g∗, λ) ≤
≤
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
eM(s[−Re λ+M−1(2b−1[−Re λ])])et Re λ dv(f, g∗, λ) ≤
by (3.4);
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1224 M. V. MARKIN
≤
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
eM(s[2α−1M−1(−Re λ)+M−1(2b−1[−Re λ])])×
× et Re λ dv(f, g∗, λ) =
by the choice: γ = max(1, 2b−1);
=
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
eM(s[2α−1+1]M−1(γ[−Re λ]))et Re λ dv(f, g∗, λ) =
by (3.6);
M(s[2α−1 + 1]M−1(γ[−Re λ])) ≤
≤ 2−NM(HNs[2α−1 + 1]M−1(γ[−Re λ])) + [1− 2−N ] ln(m0L) =
with some H > 1 and L > 0;
= (m0L)[1−2−N ]
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
e2−N M(HN s[2α−1+1]M−1(γ[−Re λ])×
× dv(f, g∗, λ)et Re λ =
by the choice: s = H−N [2α−1 + 1]−1 > 0;
= (m0L)[1−2−N ]
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
e[t−2−N γ] Re λ dv(f, g∗, λ) ≤
by the choice: 2−Nγ ≤ t;
≤ (m0L)[1−2−N ]
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
1 dv(f, g∗, λ) ≤
≤ (m0L)[1−2−N ]v(f, g∗, σ(A)) ≤
by (2.3);
≤ (m0L)[1−2−N ]4M‖f‖‖g∗‖ < ∞. (5.1)
If a > 0,
∫
{λ∈σ(A)|Re λ≤min(−M(R),a)}
es|λ|1/β
et Re λ dv(f, g∗, λ) =
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ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1225
=
∫
{λ∈σ(A)|Re λ≤min(−M(R),−a)}
es|λ|1/β
et Re λ dv(f, g∗, λ) +
+
∫
{λ∈σ(A)|min(−M(R),−a)<Re λ≤−M(R)}
es|λ|1/β
et Re λ dv(f, g∗, λ) < ∞.
Indeed, the latter integral is finite due to the boundedness of the set
{
λ ∈ σ(A)
∣∣
min(−a,−M(R)) < Re λ ≤ −M(R)
}
(note that, for a ≤ M(R), the set is, obvi-
ously, empty), the continuity of the integrated function, and the finiteness of the positive
measure v(f, g∗, ·) (see (2.3)).
The former of the two above integrals is finite as well:
∫
{λ∈σ(A)|Re λ≤min(−M(R),−a)}
eM(s|λ|)et Re λ dv(f, g∗, λ)×
×
∫
{λ∈σ(A)|Re λ≤min(−M(R),−a)}
eM(s[−Re λ+M−1(b−1(a−Re λ))])et Re λ dv(f, g∗, λ) ≤
since, for Re λ ≤ −a, a− Re λ ≤ −2 Re λ;
≤
∫
{λ∈σ(A)|Re λ≤min(−M(R),−a)}
eM(s[−Re λ+M−1(2b−1[−Re λ])])×
analogously to (5.1);
× et Re λ dv(f, g∗, λ) < ∞.
Thus, we have proved that, for an arbitrary Borel subset σ(A) ⊆ σ(A), any f ∈ X and
g∗ ∈ X∗, ∫
σ
(A)T (s|λ|)et Re λ dv(f, g∗, λ) < ∞, t > 0, (5.2)
with s = s(t) = H−N [2α−1 + 1]−1 > 0.
Furthermore, for any f ∈ X, g∗ ∈ X∗, t > 0 and s = s(t) = H−N [2α−1 +
+1]−1 > 0:
sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ(A)|T (s|λ|)et Re λ>n}
T (s|λ|)et Re λ dv(f, g∗, λ) →
→ 0 as n →∞. (5.3)
Indeed, as follows from the preceding arguments, the specific choice of
s = H−N [2α−1 + 1]−1 > 0
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1226 M. V. MARKIN
allows to partition the set
{
λ ∈ σ(A)
∣∣ T (s|λ|)et Re λ > n
}
into two Borel subsets σ1
and σ2 in such a way that σ1 is bounded and
T (s|λ|)et Re λ ≤ 1, λ ∈ σ2.
Therefore,
sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ(A)|T (s|λ|)et Re λ>n}
T (s|λ|)et Re λ dv(f, g∗, λ) ≤
≤ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ1|T (s|λ|)et Re λ>n}
T (s|λ|)et Re λ dv(f, g∗, λ) +
+ sup
{g∗∈X∗ | ‖g∗‖=1}
∫
{λ∈σ2|T (s|λ|)et Re λ>n}
T (s|λ|)et Re λ dv(f, g∗, λ) ≤
since σ1 is bounded, there is such a C > 0 that
T (s|λ|)et Re λ ≤ C, λ ∈ σ1;
by (2.4);
≤ sup
{g∗∈X∗ | ‖g∗‖=1}
C4M‖EA({λ ∈ σ1|T (s|λ|)et Re λ > n})f‖‖g∗‖ +
+ sup
{g∗∈X∗ | ‖g∗‖=1}
4M‖EA({λ ∈ σ2|T (s|λ|)et Re λ > n})f‖‖g∗‖ =
= 4CM‖EA({λ ∈ σ1|T (s|λ|)et Re λ > n})f‖ +
+ 4M‖EA({λ ∈ σ2|T (s|λ|)et Re λ > n})f‖ →
by the strong continuity of the s.m.;
→ 0 as n →∞.
According to [22], Proposition 3.1, (5.2) and (5.3) imply that, for any f ∈ X and t > 0,
etAf ∈ D(T (s|A|)),
where s = s(t) = H−N [2α−1 + 1]−1 > 0.
Hence, for any f ∈ X,
etAf ∈
⋃
s>0
D(T (s|A|)) ⊆ C{mn}(A), 0 < t < ∞.
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ON SCALAR-TYPE SPECTRAL OPERATORS AND CARLEMAN ULTRADIFFERENTIABLE ... 1227
“Only if” part. Let’s prove this part by contrapositive, i.e., we assume that for any
real a and positive b,
σ(A) \
{
λ ∈ C
∣∣ Re λ ≤ a− bM(| Im λ|)
}
6= ∅.
Therefore, for any natural n,
σ(A) \
{
λ ∈ C
∣∣∣∣ Re λ ≤ − 1
n
M(| Im λ|)
}
is unbounded.
Hence, one can choose a sequence of points in the complex plane {λn}∞n=1 in the
following way:
λn ∈ σ(A), n = 1, 2, . . . ,
Re λn > − 1
n
M(| Im λ|), n = 1, 2, . . . ,
λ0 := 0, |λn| > max
[
n, |λn−1|
]
, n = 1, 2, . . . .
The latter, in particular, implies that the points λn are distinct:
λi 6= λj , i 6= j.
Since the set {
λ ∈ C
∣∣∣∣ Re λ > − 1
n
M(| Im λ|), |λ| > max
[
n, |λn−1|
]}
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
}
,
i.e., for any λ ∈ ∆n:
Re λ > − 1
n
M(| Im λ|),
|λ| > max
[
n, |λn−1|
]
.
(5.4)
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
0 < εn <
1
n
, n = 1, 2, . . . , (5.5)
and
∆i ∩∆j = ∅, i 6= j (the disks are pairwise disjoint).
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) 6= ∅ and ∆n is open and
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1228 M. V. MARKIN
EA(∆i)EA(∆j) = 0, i 6= j. (5.6)
We can choose a unit vector en in each subspace EA(∆n)X (‖en‖ = 1) and thereby
obtain a vector sequence such that
EA(∆i)ej = δijei
(δij is Kronecker’s delta symbol).
The latter, in particular, implies that, the vectors {e1, e2, . . . } are linearly indepen-
dent. Moreover, there is an ε > 0 such that
dn := dist
(
en, span({ei | i ∈ N, i 6= n})
)
≥ ε, n = 1, 2, . . . . (5.7)
Otherwise there is a subsequence
{
dn(k)
}∞
k=1
such that dn(k) → 0 as k → ∞. Hence,
for any k = 1, 2, . . . , we can find an fn(k) ∈ span({ei|i ∈ N, i 6= n}) such that
‖en(k) − fn(k)‖ < dn(k) + 1/n(k), which immediately implies that
en(k) = EA(∆n(k))(en(k) − fn(k)) → 0 as k →∞.
Thus, the assumption that (5.7) doesn’t hold leads to a contradiction.
As follows from the Hahn – Banach Theorem, for each n = 1, 2, . . . , there is a
e∗n ∈ X∗ such that
‖e∗n‖ = 1,
〈ei, e
∗
j 〉 = δijdi.
Let
g∗ :=
∞∑
n=1
1
n2
e∗n.
Hence,
〈en, g∗〉 =
dn
n2
≥
by (5.7);
≥ ε
n2
. (5.8)
Concerning the sequence of the real parts, {Re λn}∞n=1, there are two possibilities: it is
either bounded, or not. Let’s consider separately each of them.
First, assume that the sequence {Re λn}∞n=1 is bounded, i.e., there is such an ω > 0
that
|Re λn| ≤ ω, n = 1, 2, . . . . (5.9)
Let
f :=
∞∑
n=1
1
n2
en.
As can be easily deduced from the (5.6),
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EA(∆n)f =
1
n2
en, n = 1, 2, . . . ,
EA(∪∞n=1∆n)f = f.
(5.10)
Also, for n = 1, 2, . . . ,
v(f, g∗,∆n) ≥ |〈EA(∆n)f, g∗〉| =
by (5.10);
=
∣∣∣∣〈 1
n2
en, g∗
〉∣∣∣∣ =
by (5.8);
=
dn
n4
≥ ε
n4
. (5.11)
For an arbitrary s > 0, we have∫
σ(A)
T (s|λ|)eRe λ dv(f, g∗, λ) =
by (5.10);
=
∫
σ(A)
T (s|λ|)eRe λ dv(EA(∪∞n=1∆n)f, g∗, λ) =
by the properties of the o.c.;
=
∫
∪∞n=1∆n
T (s|λ|)eRe λ dv(f, g∗, λ) =
=
∞∑
n=1
∫
∆n
T (s|λ|)eRe λ dv(f, g∗, λ) ≥
for λ ∈ ∆n, by (5.4), (5.9), and (5.5): |λ| ≥ n, and Re λ = Re λn
−(Re λn − Re λ) ≥ Re λn − |λn − λ| ≥ −ω − εn ≥ −ω − 1;
≥
∞∑
n=1
T (sn)e−(ω+1)v(f, g∗,∆n) ≥
by (5.11);
≥ e−(ω+1)
∞∑
n=1
εT (sn)
n4
= ∞.
Indeed, by definition (3.1)
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1230 M. V. MARKIN
T (sn) ≥ m0
(sn)4
m4
, n = 1, 2, . . . .
Thus, by [22], Proposition 3.1,
eAf 6∈
⋃
t>0
D(T (t|A|)).
Then, by (3.3),
eAf 6∈ C{mn}(A).
Hence, according to Proposition 4.1, the orbit etAf, t ≥ 0, does not belong to
C{mn}
(
(0,∞), X
)
.
Now, suppose that the sequence {Re λn}∞n=1 is unbounded. The sequence being
bounded above, since A generates a C0-semigroup [3] (see also [1]), this means there is
a subsequence {Re λn(k)}∞k=1 such that
Re λn(k) ≤ −k, k = 1, 2, . . . . (5.12)
Consider the vector
f :=
∞∑
k=1
1
k2
en(k).
By (5.6),
EA(∆n(k))f =
1
k
en(k), k = 1, 2, . . . ,
EA(∪∞k=1∆n(k))f = f.
For an arbitrary s > 0, we have similarly∫
σ(A)
T (s|λ|)eRe λ dv(f, g∗, λ) =
∞∑
k=1
∫
∆n(k)
T (s|λ|)eRe λ dv(f, g∗, λ) = ∞.
Indeed, for all λ ∈ ∆n(k), based on (5.5), (5.12), and (5.4), we have
Re λ = Re λn(k) − (Re λn(k) − Re λ) ≤ Re λn(k) + |λn(k) − λ| ≤
≤ Re λn(k) + εn(k) ≤ −k + 1 ≤ 0
and
− 1
n(k)
M(| Im λ|) < Re λ.
Therefore, for λ ∈ ∆n(k):
− 1
n(k)
M(| Im λ|) < Re λ ≤ −k + 1 ≤ 0.
Whence, for λ ∈ ∆n(k),
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Re λ ≤ −k + 1 ≤ 0 and |λ| ≥ | Im λ| ≥ M−1(n(k)[−Re λ]).
Using these estimates we have∫
∆n(k)
T (s|λ|)eRe λ dv(f, g∗, λ) ≥
∫
∆n(k)
eM(s|λ|)eRe λ dv(f, g∗, λ) ≥
≥
∫
∆n(k)
eM(sM−1(n(k)[−Re λ]))eRe λ dv(f, g∗, λ) ≥
by (3.5), M(λ) ≥ 2−nM(hnλ) + [1− 2−n] ln(m0l), λ ≥ 0, n = 1, 2, . . . ;
with some h > 1 and l > 0;
for a sufficiently large natural N so that hNs ≤ 1;
≥ (m0l)[1−2N ]
∫
∆n(k)
e2−N M(hN sM−1(n(k)[−Re λ]))eRe λ dv(f, g∗, λ) ≥
≥ (m0l)[1−2N ]
∫
∆n(k)
e2−N M(M−1(n(k)[−Re λ]))eRe λ dv(f, g∗, λ) ≥
≥ (m0l)[1−2N ]
∫
∆n(k)
e(2−N n(k)−1)[−Re λ] dv(f, g∗, λ) ≥
for all k’s sufficiently large so that 2−Nn(k)− 1 > 0 and k − 1 ≥ 1;
≥ (m0l)[1−2N ]e[2−N n(k)−1](k−1)v(f, g∗,∆n(k)) ≥
by (5.11);
≥ (m0l)[1−2N ] εe
2−N n(k)−1
n(k)4
→∞ as k →∞.
Similarly to the above, we infer that the orbit etAf, t ≥ 0, does not belong to the class
C{mn}
(
(0,∞), X
)
.
Thus, all the possibilities concerning {Re λn}∞n=1 analyzed, the “only if” part has
been proved by contrapositive.
The theorem is proved.
In particular, for mn = [n!]β , 1 ≤ β < ∞, we obtain Theorem 5.1 of [2].
As well as in (3.3), the function T (·) in Theorem 5.1 is replaceable by any nonneg-
ative, continuous, and increasing function L(·) defined on [0,∞) such that
c1L(γ1λ) ≤ T (λ) ≤ c2L(γ2λ), λ > R,
with some positive γ1, γ2, c1, c2, and a nonnegative R.
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1232 M. V. MARKIN
6. Acknowledgements. I’m eternally grateful to my mother Svetlana A. Markina,
whose love, constant support and unsurpassed patience made this humble dedication
possible.
I’d also like to express my highest appreciation (long overdue) to Mrs. Lilia Zusman,
one of the kindest human beings I’ve ever known.
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Received 23.01.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 9
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