A NOTE ON THE SPECTRAL OPERATORS OF SCALAR TYPE AND ...downloads.hindawi.com/journals/ijmms/2002/795412.pdf · On account of compactness, the terms spectral measure and operational

IJMMS 32:10 (2002) 635–640PII. S0161171202112221

http://ijmms.hindawi.com© Hindawi Publishing Corp.

A NOTE ON THE SPECTRAL OPERATORS OF SCALAR TYPEAND SEMIGROUPS OF BOUNDED LINEAR OPERATORS

MARAT V. MARKIN

Received 20 December 2001

It is shown that, for the spectral operators of scalar type, the well-known characteriza-tions of the generation of C0- and analytic semigroups of bounded linear operators canbe reformulated exclusively in terms of the spectrum of such operators, the conditions onthe resolvent of the generator being automatically met and the corresponding semigroupbeing that of the exponentials of the operator.

2000 Mathematics Subject Classification: 47B40, 47D03, 34G10, 47B15, 47D06.

1. Introduction. As known, the celebrated criteria of the generation of C0- and an-

alytic semigroups of bounded linear operators contain conditions on the geometry of

the spectrum of the generator along with rather stringent restrictions on its resolvent

behavior [6, 7, 12, 15].

For a normal operator in a complex Hilbert space, the restrictions on the resolvent

can be dropped, being automatically satisfied when the conditions on the spectrum

of the generator are met [6, 13].

If A is such an operator and EA(·) is its spectral measure (resolution of the identity),

the generated semigroup consists of its exponentials in the sense of the corresponding

operational calculus [4, 13]

etA =∫σ(A)

etλdEA(λ), t ≥ 0. (1.1)

It is the purpose of the present note to highlight the fact that the criteria acquire

purely geometrical form in the more general case of a spectral operator of scalar type

(scalar operators) in a complex Banach space [2, 5].

Observe for that matter that, in a Hilbert space, the scalar operators are the opera-

tors similar to normal ones [14].

2. Preliminaries. Henceforth, unless specifically stated otherwise, A is a scalar op-

erator in a complex Banach space X with a norm ‖·‖ and EA(·) is its spectral measure.

For such operators, there is an operational calculus for Borel measurable functions

on the spectrum [2, 5].

If F(·) is a Borel measurable function on the spectrum of A, σ(A), a new scalar

operator

F(A)=∫σ(A)

F(λ)dEA(λ) (2.1)

http://ijmms.hindawi.com

http://ijmms.hindawi.com

http://www.hindawi.com

636 MARAT V. MARKIN

is defined as follows:

F(A)f := limn→∞Fn(A)f , f ∈D(F(A)),

D(F(A)

):={f ∈X∣∣ lim

n→∞Fn(A)f exists},

(2.2)

where

Fn(·) := F(·)χ{λ∈σ(A)||F(λ)|≤n}(·), n= 1,2, . . . , (2.3)

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

Fn(A) :=∫σ(A)

Fn(λ)dEA(λ), n= 1,2, . . . , (2.4)

being the integrals of bounded Borel measurable functions on σ(A), are bounded

scalar operators on X defined in the same way as for normal operators (see, e.g.,

[4, 13]).

The properties of the spectral measure EA(·) and the operational calculus, which

underlie the entire argument henceforth, are exhaustively delineated in [2, 5]. We just

note here that, due to its strong countable additivity, the spectral measure EA(·) is

bounded, that is, there is an M > 0 such that

∥∥EA(δ)∥∥≤M, for any Borel set δ. (2.5)

Observe that, in (2.5), the same notation as for the norm in X, ‖ · ‖, was used to

designate the norm in the space of bounded linear operators on X, �(X). We will also

adhere to this rather common economy of symbols in what follows for the norm in

the dual space X∗.

On account of compactness, the terms spectral measure and operational calculus

for spectral operators, repeatedly referred to, will be abbreviated to s.m. and o.c.,

respectively.

3. C0-semigroups

Proposition 3.1. A scalar operator A in a complex Banach space X generates a

C0-semigroup of bounded linear operators if and only if, for some real ω,

Reλ≤ω, λ∈ σ(A), (3.1)

in which case the semigroup consists of the exponentials

etA =∫σ(A)

etλdEA(λ), t ≥ 0. (3.2)

Proof. Condition (3.1), being a constituent of the general C0-semigroup genera-

tion criterion [6, 7, 15], obviously remains necessary. Thus, we are only to prove its

sufficiency.

From (3.1), we immediately infer that {λ∈ C | Reλ >ω} ⊆ ρ(A), where ρ(A) is the

resolvent set of A.

ON SCALAR OPERATORS AND SEMIGROUPS 637

Furthermore, for z >ω, we have

∥∥R(z,A)n∥∥=∥∥∥∥∫σ(A)

(z−λ)−ndEA(λ)∥∥∥∥ by the properties of the o.c.

≤ 4M supλ∈σ(A)

|z−λ|−n where M is a constant from (2.5)

= 4M[dist

(z,σ(A)

)]−nby (3.1)

≤ 4M(z−ω)n .

(3.3)

Whence, by the C0-semigroup generation criterion (see, e.g., [6]), it follows that Agenerates a certain C0-semigroup of bounded linear operators {T(t) | t ≥ 0}.

From condition (3.1), we also obtain the estimate

∣∣etλ∣∣= etReλ ≤ eωt, t ≥ 0, λ∈ σ(A), (3.4)

which, by the properties of the o.c., implies that exponentials (3.2) form a semigroup

of bounded linear operators.

To prove that {etA | t ≥ 0} is a C0-semigroup, it is enough to demonstrate its conti-

nuity at 0 in the weak sense [15].

For any f ∈X and an arbitrary g∗ ∈X∗,

∣∣⟨etAf −f ,g∗⟩∣∣=∣∣∣∣∫σ(A)

(etλ−1

)d⟨EA(λ)f ,g∗

⟩∣∣∣∣≤∫σ(A)

∣∣etλ−1∣∣dv(f ,g∗,λ), (3.5)

by the properties of the o.c., 〈·,·〉 being the pairing between the space X and its

dual, X∗, where v(f ,g∗,·), the total variation of the complex-valued Borel measure

〈EA(·)f ,g∗〉, is a finite positive Borel measure.

By the Lebesgue Dominated Convergence theorem, whose conditions as easily seen

are readily met, the latter integral approaches 0 as t→ 0.

Finally, we need to prove that T(t)= etA, t ≥ 0.

We first show that, for arbitrary f ∈D(A) and g∗ ∈X∗,

ddt⟨etA,f∗

⟩= ⟨AetAf ,g∗⟩, t ≥ 0. (3.6)

Note that, by the properties of the o.c., for all f ∈D(A),

eAtf ∈D(A), AetAf = etAAf , t ≥ 0. (3.7)

Fix a t ≥ 0 and choose a small segment [a,b] ⊂ [0,∞) so that t is its left endpoint

if t = 0 and the midpoint otherwise. For all sufficiently small increments ∆t ≠ 0 such

638 MARAT V. MARKIN

that t+∆t ∈ [a,b], and arbitrary f ∈D(A) and g∗ ∈X∗, we have

∣∣∣∣∣⟨e(t+∆t)Af −etAf

∆t−AetAf ,g∗

⟩∣∣∣∣∣ by the properties of the o.c.

=∣∣∣∣∣∫σ(A)

[e(t+∆t)λ−etλ

∆t−λetλ

]d⟨EA(λ)f ,g∗

⟩∣∣∣∣∣≤∫σ(A)

∣∣∣∣∣e(t+∆t)λ−etλ

∆t−λetλ

∣∣∣∣∣dv(f ,g∗,λ) �→ 0 as ∆t �→ 0,

(3.8)

by the Lebesgue Dominated Convergence theorem,

∣∣∣∣∣(e(t+∆t)λ−etλ)

∆t−λetλ

∣∣∣∣∣ �→ 0 as ∆t �→ 0, λ∈ σ(A). (3.9)

For λ∈ σ(A),∣∣∣∣∣e

(t+∆t)λ−etλ∆t

−λetλ∣∣∣∣∣ by the total change theorem

≤ 2 maxa≤s≤b

∣∣λesλ∣∣= 2 maxa≤s≤b

esReλ|λ| by (3.1)

≤ 2 maxa≤s≤b

esω|λ| without restricting generality, ω can be regarded to be positive

≤ 2ebω|λ|.(3.10)

Since f ∈ D(A), ∫σ(A) |λ|dv(f ,g∗,λ) < ∞ for any g∗ ∈ X∗ [3, 5]. Thus, for any

f ∈D(A) and g∗ ∈D(A∗), where A∗ is the adjoint of A (D(A) is dense in X),

ddt⟨etAf ,g∗

⟩= ⟨AetAf ,g∗⟩= ⟨etAf ,A∗g∗⟩, t ≥ 0, (3.11)

that is, e·Af is a weak solution of the equation

y ′(t)=Ay(t) (3.12)

on [0,∞) in the sense of [1].

Since A generates the C0-semigroup {T(t) | t ≥ 0}, such a solution is unique for any

f ∈X and is the orbit T(t)f , t ≥ 0 [1].

Therefore, for any f ∈D(A): etAf = T(t)f , t ≥ 0. Whence, by the density of D(A)in X, we conclude that

etA = T(t), t ≥ 0. (3.13)

ON SCALAR OPERATORS AND SEMIGROUPS 639

4. Analytic semigroups

Proposition 4.1. A scalar operator A in a complex Banach space X generates an

analytic semigroup of bounded linear operators if and only if, for some real ω and

0< θ <π/2,

σ(A)⊆ {z ∈ C ∣∣ ∣∣arg(z−ω)∣∣≥π/2+θ}, (4.1)

where arg· is the principal value of the argument from the interval (−π,π].The semigroup is analytically continuable into the sector Σθ = {z ∈ C | |argz| < θ}

by the formula

ezA =∫σ(A)

ezλdEA(λ), z ∈ Σθ. (4.2)

Proof. Condition (4.1) is necessary since it is a constituent of the general criterion

of generation of analytic semigroup [6, 7, 15]. Thus, we need to validate its sufficiency

only.

First, we infer that ρ(A)⊆ {z ∈ C | |arg(z−ω)|<π/2+θ}.For an arbitrary 0< ε < θ and any z such that |arg(z−ω)|<π/2+θ−ε, there are

two possibilities:

(a) π/2−θ < |arg(z−ω)| ≤π/2+θ−ε,(b) |arg(z−ω)| ≤π/2−θ.

In the first case,

∥∥R(z,A)∥∥=∥∥∥∥∫σ(A)

(z−λ)−1dEA(λ)∥∥∥∥ by the properties of the o.c.


|z−λ|−1 where M is a constant from (2.5)

= 4M[dist

(z,σ(A)

)]−1 ≤ 4M|z−ω|sinε

.

(4.3)

In the second case,

∥∥R(z,A)∥∥=∥∥∥∥∫σ(A)

(z−λ)−1dEA(λ)∥∥∥∥ by the properties of the o.c.


|z−λ|−1 = 4M[dist

(z,σ(A)

)]−1 ≤ 4M|z−ω| .

(4.4)

Thus, for any 0< ε < θ,

∥∥R(z,A)∥∥≤ 4M cscε|z−ω| whenever

∣∣arg(z−ω)∣∣< π2+θ−ε. (4.5)

Condition (4.1) and the latter estimate imply thatA generates an analytic semigroup

{T(t) | t ≥ 0} (see, e.g., [6]).

The fact that the exponentials

ezA =∫σ(A)

ezλdEA(λ), |argz|< θ, (4.6)

640 MARAT V. MARKIN

are bounded linear operators with the semigroup property can be easily inferred from

(4.1). And the fact that etA = T(t), t ≥ 0, can be substantiated in the same way as in

the case of C0-semigroups above.

5. Some final remarks. The author intentionally did not entertain the idea of de-

veloping here similar arguments for differentiable C0-semigroups [6, 12] and, for that

matter, the C0-semigroups with orbits belonging to the Gevrey or, more generally,

Carleman classes of ultradifferentiable vector functions leaving that for a discussion

in a more general context (for the case of the normal operators, see [8, 9, 10, 11]).

References

[1] J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constantsformula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370–373.

[2] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958),217–274.

[3] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and AppliedMathematics, vol. 7, Interscience Publishers, New York, 1958.

[4] , Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space,Interscience Publishers, New York, 1963.

[5] , Linear Operators. Part III: Spectral Operators, Pure and Applied Mathematics,vol. 7, Interscience Publishers, New York, 1971.

[6] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad-uate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000.

[7] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American MathematicalSociety Colloquium Publications, vol. 31, American Mathematical Society, RhodeIsland, 1957.

[8] M. V. Markin, On the ultradifferentiability of weak solutions of a first-order operator-differential equation in Hilbert space, Dopov. Nats. Akad. Nauk Ukraïni (1996), no. 6,22–26.

[9] , On the strong smoothness of weak solutions of an abstract evolution equation. I.Differentiability, Appl. Anal. 73 (1999), no. 3-4, 573–606.

[10] , On the strong smoothness of weak solutions of an abstract evolution equation. II.Gevrey ultradifferentiability, Appl. Anal. 78 (2001), no. 1-2, 97–137.

[11] , On the strong smoothness of weak solutions of an abstract evolution equation. III.Gevrey ultradifferentiability of orders less than one, Appl. Anal. 78 (2001), no. 1-2,139–152.

[12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.

[13] A. I. Plesner, Spectral Theory of Linear Operators, Nauka, Moscow, 1965 (Russian).[14] J. Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355–

361.[15] K. Yosida, Functional Analysis, Die Grundlehren der Mathematischen Wissenschaften,

vol. 123, Academic Press, New York, 1965.

Marat V. Markin: Department of Mathematics and Statistics, Boston University,111 Cummington Street, Boston, MA 02215, USA

E-mail address: [email protected]

mailto:[email protected]

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Documents

A NOTE ON THE SPECTRAL OPERATORS OF SCALAR TYPE AND ...downloads.hindawi.com/journals/ijmms/2002/795412.pdf · On account of compactness, the terms spectral measure and operational