On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

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Volume 2011, Article ID 825951,27pages doi:10.1155/2011/825951

Research Article

On the Differentiability of Weak Solutions of

an Abstract Evolution Equation with a Scalar Type

Spectral Operator

Marat V. Markin

Department of Partial Diﬀerential Equations, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchekivs’ka Street, 01601 Kiev, Ukraine

Correspondence should be addressed to Marat V. Markin,[email protected]

Received 11 December 2010; Accepted 15 February 2011

Academic Editor: Palle E. Jorgensen

Copyrightq2011 Marat V. Markin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For the evolution equationyt Aytwith a scalar type spectral operatorAin a Banach space, conditions onAare found that are necessary and sufficient for all weak solutions of the equation on0,∞to be strongly infinite differentiable on0,∞or0,∞. Certain effects of smoothness improvement of the weak solutions are analyzed.

1. Introduction

Consider the evolution equation

yt Ayt 1.1

with a scalar type spectral operatorAin a complex Banach spaceX.

Following1, we understand by a weak solution of equation1.1on an interval0, T

0< T ≤ ∞such a vector functiony:0, T → Xthat

1y·is strongly continuous on0, T;

2for anyg∗∈DA∗,

d dt

yt, g∗yt, A∗g∗, t∈0, T. 1.2

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As was shown2, Theorem4.2, the general weak solution of1.1on0, T 0< T≤ ∞

is as follows:

yt etAf, t∈0, T, f ∈

0≤t<T

DetA, _1.3

the operator exponentialsetA_{, 0} _≤ _{t < T}_{, defined in the sense of the operational calculus for}

scalar type spectral operatorssee Section2.

Here, we are to find conditions on A that are necessary and suﬃcient for all weak

solutions of1.1on0,∞to be strongly infinite diﬀerentiable on0,∞or0,∞.

This goal attained; all the principal results of paper3and the corresponding ones of

4obtain their natural generalizations.

2. Preliminaries

Henceforth, unless specified otherwise, let Abe a scalar type spectral operator in a complex Banach spaceXwith a norm · ,EA·its spectral measurethe resolution of the identity, and σAthe operator’s spectrum, with the latter being the support for the former.

Observe that, in a Hilbert space, the scalar type spectral operators are those similar to the

normal ones5.

For scalar type spectral operators, there has been developed an operational calculus for Borel measurable functions defined onσA 6,7. WithF·being such a function, a new scalar type spectral operator

FAdef

σAFλdEAλ 2.1

is defined as follows:

FAfdef lim

n→ ∞FnAf, f∈DFA,

DFAdef

f∈X| lim

n→ ∞FnAf exists ,

2.2

where

Fn·def F·χ{λ∈σA||Fλ|≤n}·, n1,2, . . . , 2.3

χα·is the characteristic function of a setαand

FnAdef

σAFnλdEAλ, n1,2, . . . , 2.4

are bounded scalar type spectral operators onX defined in the same manner as for normal

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In particular,

A

σAλ dEAλ. 2.5

The properties of the spectral measure, EA·, and the operational calculus underlying our discourse are exhaustively delineated in6,7. We will outline here a few noteworthy facts.

Due to its strong countable additivity, the spectral measureEA·is bounded10, that

is, there is anM >0 such that

EAα ≤M, α∈ B 2.6

Bis theσ-algebra of Borel sets in the complex plane .

Observe that · is used in2.6to designate the norm in the space of bounded linear operators on X. We will adhere to this rather common economy of symbols adopting the same notation for the norm in the dual spaceX∗as well.

For anyf ∈X andg∗ ∈ X∗, letvf, g∗,·be the total variation of the complex-valued measureEA·f, g∗ onB.

As we discussed in4,11, withF·being an arbitrary Borel measurable function on , for anyf ∈DFA,g∗∈X∗, andδ∈ B,

δ

|Fλ|dvf, g∗, λ≤4MEAδFAfg∗, 2.7

whereM >0 is the constant from2.6.

In particular, forFλ 1,λ ∈ , andδ or any Borel set containingσA, we have

vf, g∗, δ≤4Mfg∗, f∈X, g∗∈X∗. 2.8

Further, for a Borel measurable nonnegative functionF· on , aδ ∈ B, and a sequence

{Δn}∞_n1of pairwise disjoint Borel sets in ,

δ

FλdvEA∪∞_n1Δn

f, g∗, λ

∞

n1

δ∩Δn

FλdvEAΔnf, g∗, λ

, f ∈X, g∗∈X∗.

2.9

Indeed, since for the spectral measure6,7

EAαEA

βEA

α∩β, α, β∈ B, 2.10

for the total variation, we have

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Whence, due to nonnegativity ofF· 12,

δ

FλdvEA

∪∞

n1Δn

f, g∗, λ

δ∩∪∞_n1Δn

Fλdvf, g∗, λ

∞

n1

δ∩Δn

Fλdvf, g∗, λ

∞

n1

δ∩ΔnFλdv

EAΔnf, g∗, λ

.

2.12

We shall need the following regions in the complex plane:

Lbdef {λ∈ |Reλ≥max0, bln|Imλ|},

Lb−,bdef {λ∈ |Reλ≤min0,−b−ln|Imλ| or Reλ≥max0, bln|Imλ|},

2.13

wherebandb−are positive constants.

The terms spectral measure and operational calculus frequently referred to will be ab-breviated to s.m. and o.c., respectively.

3. Differentiability of a Particular Weak Solution

Proposition 3.1. Letn1,2, . . .andI be a subinterval of an interval0, T 0< T ≤ ∞. A weak

solutiony·of 1.1on0, Tisntimes strongly diﬀerentiable onIif and only if

yt∈DAn, t∈I, 3.1

in which case,

ykt Akyt, k1,2, . . . , n, t∈I. 3.2

Proof. “Only if” part.

Let n 1,2, . . .and suppose that a weak solution y·of 1.1 on 0, T isn times

strongly diﬀerentiable on a subintervalI⊆0, T. Then for anyg∗∈DA∗,

yt, g∗ d

dt

yt, g∗yt, A∗g∗, t∈I. 3.3

Hence, by the closedness of the operatorAcf.1,

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This concludes proving the “only if” part for n 1, in which case, as is to be noted, the subintervalIcan shrink to a single pointt∈0, T.

Let n ≥ 2 and let the interval I be not a singleton. Then diﬀerentiating 3.4 at an arbitraryt∈I, we obtain

yt lim

Δt→0

yt Δt−yt

Δt Δtlim→0A

yt Δt−yt

Δt , 3.5

with the incrementsΔtbeing such thatt Δ∈I. Since

lim

Δt→0

yt Δt−yt

Δt y

_t_, _t_∈_I, _3.6

by the closedness ofA, we infer that

yt∈DA, yt Ayt, t∈I. 3.7

Considering3.4and3.7,

yt∈DA2, yt A2yt, t∈I. 3.8

Continuing inductively in this manner, we arrive at

yt∈DAk, ykt Akyt, k1,2, . . . , n, t∈I. 3.9

“If” part.

Lety·be a weak solution of1.1on an interval0, T 0 < T ≤ ∞such that for an

n1,2, . . .and a subintervalI ⊆0, T,

yt∈DAn, t∈I. 3.10

As was discussedsee Section1,

yt etAf, t∈0, T, 3.11

with somef∈₀_≤_t<TDetA_.

The fact that etA_f _∈ _D_An_, _t _∈ _I_{, implies by the properties of the o.c. and} ₂_,

Proposition3.1, that, for anyg∗∈X∗,

σA|λ|

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Given ak1, . . . , nand an arbitraryt∈I, let us choose a segmenta, b⊂Ia < bso thatt

is its midpoint if 0< t < Tora0 ift0. For incrementsΔtsuch thata≤t Δt≤band any

g∗∈X∗, we have

Ak−1yt Δt−Ak−1yt

Δt −A

k_y_t_{, g}∗

by the properties of theo.c.;

σA

λk−1_etΔtλ₋_λk−1_etλ

Δt −λ

k_etλ

dEAλf, g∗

σA

Δt −λ

k_etλ

dEAλf, g∗

≤ σA

Δt −λ

k_etλ

dv

f, g∗, λ

by theLebesgue Dominated Convergence Theorem;

−→0 asΔt−→0.

3.13

Indeed, for anyk1, . . . , nand an arbitraryλ∈σA,

Δt −λ

k_etλ

≤λk−1etΔtλ_Δ−λk−1etλ

t

λketλ by theTotal Change Theorem;

≤2|λ|kmax

a≤s≤be

sReλ_≤₂

⎧ ⎪ ⎨ ⎪ ⎩

|λ|keaReλ _{if Re}_{λ <}₀_,

|λ|kebReλ _{if Re}_λ_≥₀_,

by3.12, considering thata, b∈I;

∈ L1_σ_A_{, v}_{f, g}∗_,_·_,

Δt −λ

k_etλ

−→0 asΔt → 0.

3.14

Therefore, for anyg∗∈X∗,

dk

dtk

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ForΔm :{λ∈ | |λ| ≤m},m1,2, . . ., let us fix an arbitraryk1, . . . , nand consider the

sequence of vector functions

ymt:EAΔmAkyt EAΔmAketAf, m1,2, . . . , t∈I. 3.16

Sincef ∈DAk_etA_,_t_∈_I_{, by the properties of the o.c.,}

EAΔmAketAf AEAΔmketAEAΔmf, m1,2, . . . , t∈I. 3.17

Due to the boundedness ofΔm,m 1,2, . . ., by the properties of the o.c.,AEAΔm,m

1,2, . . ., is a bounded linear operator onXAEAΔm ≤4Mm,m1,2, . . .,7. Hence, the

vector functions

ymt EAΔmAketAf, m1,2, . . . , t∈I, 3.18

are strongly continuous onI.

For an arbitrary segmenta, b⊆I, we have

sup

a≤t≤b

AketAf−EAΔmAketAf by the properties of theo.c.;

sup

a≤t≤b

{λ∈σA||λ|>m}λ

k_etλ_dE

Aλf

as follows from theHahn-Banach Theorem;

sup

a≤t≤b

sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA||λ|>m}λ

k_etλ_dEA_λ_{f, g}∗

sup

a≤t≤b

sup

{g∗∈X∗|g∗1}

{λ∈σA||λ|>m}λ

k_etλ_d_EA_λ_{f, g}∗

≤ sup

a≤t≤b

sup

{g∗∈X∗|g∗1}

{λ∈σA||λ|>m}|λ|

k_etReλ_dv_{f, g}∗_{, λ}

≤ sup

{g∗_∈_X∗_|_g∗₁}

sup

a≤t≤b

{λ∈σA||λ|>m,Reλ≤0}|λ|

k_etReλ_dv_{f, g}∗_{, λ}

{λ∈σA||λ|>m,Reλ>0}|λ|

k_etReλ_dv_{f, g}∗_{, λ}

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≤ sup

{g∗∈X∗|g∗1}

{λ∈σA||λ|>m,Reλ≤0}|λ|

k_eaReλ_dv_{f, g}∗_{, λ}

sup

{g∗∈X∗|g∗1}

{λ∈σA||λ|>m,Reλ>0}|λ|

m_ebReλ_dv_{f, g}∗_{, λ}

≤ sup

{g∗∈X∗|g∗1}

k_eaReλ_dv_{f, g}∗_{, λ}

sup

{g∗_∈_X∗_|_g∗₁}

k_ebReλ_dv_{f, g}∗_{, λ} _by₂_.₇_;

≤4M

⎡

⎣ sup

{g∗∈X∗|g∗1}

EA{λ∈σA| |λ|> m}AkeaAfg∗

sup

{g∗∈X∗|g∗1}

EA{λ∈σA| |λ|> m}AkebAfg∗

⎤ ⎦

4MEA{λ∈σA| |λ|> m}AkeaAfEA{λ∈σA| |λ|> m}AkebAf

by the strong continuity of thes.m.;

−→0 asm−→ ∞.

3.19

Hence, for any k 1, . . . , n, the vector function Ak_etA_f_, _t _∈ _I_{, is strongly continuous on} I, being the uniform limit of the sequence of strongly continuous on I vector functions

{ym·}∞m1on any segmenta, b⊆I.

Let us fix an arbitrarya∈Iand integrate3.15fork1 betweenaand an arbitrary

t∈I. Considering the strong continuity ofAetA_f_,_t_∈_I_{, we have}

etAf−eaAf, g∗

t

a

AesAf ds, g∗

, g∗∈X∗. 3.20

Whence, as follows from the Hahn-Banach Theorem,

etAf−eaAf

t

a

AesAf ds, t∈I. 3.21

By the strong continuity ofAetA_f_,_t_∈_I_,

d

dte

tA_f _AetA_f, _t_∈_I. _3.22

Consequently, by3.15, forn2,

d dt

d

dte

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Whence, analogously,

d2

dt2e

tA_f _A2_etA_f, _t_∈_I. _3.24

Continuing inductively in this manner, we infer that, for anyk1, . . . , n,

dk

dtke

tA_f _Ak_etA_f, _t_∈_I. _3.25

Corollary 3.2. LetIbe a subinterval of an interval0, T 0< T ≤ ∞. A weak solutiony·of1.1

on0, Tis strongly infinite diﬀerentiable onIif and only if

yt∈C∞Adef

∞

n1

DAn, t∈I, 3.26

in which case,

ynt Anyt, n1,2, . . . , t∈I. 3.27

Thus, we have obtained generalizations of Proposition 4.1 and Corollary 4.1 of3, respec-tively.

4. Differentiability of Weak Solutions

Theorem 4.1. Every weak solution of 1.1on0,∞is strongly infinite diﬀerentiable on0,∞if

and only if there is ab >0 such that the setσA\ Lbis bounded.

Proof. “If” part.

Letb >0 be such that the setσA\ Lbis bounded andy·a weak solution of1.1

on0,∞. Thensee Section1

yt etAf, 0≤t <∞, 4.1

with somef∈₀_≤_t<_∞DetA_.

For anyn1,2, . . .,t≥0 and an arbitraryg∗∈X∗,

σA|λ|

n_etReλ_dv_{f, g}∗_{, λ}

σA\Lb

|λ|netReλdvf, g∗, λ

σA∩Lb

|λ|netReλdvf, g∗, λ<∞.

4.2

Indeed, the former integral is finite due to the boundedness of the setσA\Lb, the finiteness

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For the latter one, we have

σA∩Lb

≤

σA∩Lb

|Reλ||Imλ|netReλdvf, g∗, λ forλ∈σA∩ Lb,Reλ≥0, |Imλ|≤eb −1

Reλ_;

≤

σA∩Lb

Reλeb−1Reλn_etReλ_dv_{f, g}∗_{, λ} _since _x≤_ex_, _x≥_0;

≤

σA∩Lb

beb−1Reλeb−1 Reλ

n

etReλdvf, g∗, λ

b1n

σAe

nb−1

tReλ_dv_{f, g}∗_{, λ} _since _f∈

0≤t<∞

DetA, by2,Proposition 3.1;

<∞.

4.3

Further, for anyn1,2, . . .andt≥0,

sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA||λ|n_etReλ_>m}|

λ|netReλdvf, g∗, λ

≤ sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA\Lb||λ|netReλ>m}

sup

{g∗∈X∗|g∗1}

{λ∈σA∩Lb||λ|netReλ>m}

−→0 asm−→ ∞.

4.4

Indeed, since, due to the boundedness of the setσA\Lb, the set{λ∈σA\Lb | |λ|netReλ> m}is void for all suﬃciently largem’s,

sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA\Lb||λ|netReλ>m}

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In addition to this

sup

{g∗_∈_X∗_|_g∗₁}

|λ|netReλdvf, g∗, λ analogously to4.3;

≤ sup

{g∗∈X∗|g∗1}

b1n

enb−1tReλ_dv_{f, g}∗_{, λ}

sincef ∈

0≤t<∞

DetA, by2.7;

≤b1n sup

{g∗∈X∗|g∗1}

4MEA"λ∈σA∩ Lb| |λ|netReλ> m

#

enb−1tA_f_g∗

4Mb1nEA

"

λ∈σA∩ Lb| |λ|netReλ> m

#

enb−1tA_f

−→0 asm−→ ∞.

4.6

By the properties of the o.c. and2, Proposition3.1,4.2and4.4imply that

yt etA_f_∈_C∞_A_, ₀_≤_{t <}_∞_. _4.7

Then, by Corollary3.2,y·is strongly infinite diﬀerentiable on0,∞. “Only if” part.

We will prove this part by contrapositive.

Assume that, for anyb>0, the setσA\ Lbis unbounded.

In particular, for anyn1,2, . . ., unbounded is the setσA\ L_2n−1.

Hence, we can choose a sequence of points in the complex plane{λn}∞n1as follows:

λn∈σA, n1,2, . . . ,

Reλn<max

0,2n−1ln|Imλ|, n1,2, . . . ,

λ0:0,|λn|>maxn4,|λn−1|

, n1,2, . . . .

4.8

The latter implies in particular that the pointsλnare distinctλi/λj,i /j.

Since, for anyn1,2, . . ., the set

"

λ∈ |Reλ <max0,2n−1ln|Imλ|, |λ|>maxn4,|λn−1|

#

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is open in , there exists anεn>0 such that this set along with the pointλncontains the open

disk of radiusεncentered atλn

Δn:{λ∈ | |λ−λn|< εn}. 4.10

Then, for anyλ∈Δn,n1,2, . . .,

Reλ <max0,2n−1ln|Imλ|, |λ|>maxn4,|λn−1|

. 4.11

Further, since the pointsλn,n1,2, . . ., are distinct, we can regard the radii of the disks,εn,

n1,2, . . ., to be small enough so that

0< εn <

1

n, n1,2, . . . ,

Δi∩Δj∅, i /j

i.e.,the disks arepairwise disjoint.

4.12

Whence, by the properties of the s.m.,

EAΔiEA

Δj

0, i /j. 4.13

Observe also, that the subspacesEAΔnX,n1,2, . . ., are nontrivial sinceΔn∩σA/∅, with

Δnbeing an open set.

By choosing a unit vectoren ∈ EAΔnX en 1,n 1,2, . . ., we obtain a vector

sequence such that

EAΔiejδijei, i, j1,2, . . . 4.14

δijis the Kronecker delta symbol.

As is readily verified, 4.14 implies that the vectors {e1, e2, . . .} are linearly independent.

Moreover, there is anε >0 such that

dn:dist

en,span{ek|k1,2, . . . , k /n}

≥ε, n1,2, . . . . 4.15

Indeed, the opposite implies the existence of a subsequence{dnk}∞k1such that

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Then, for anyk 1,2, . . ., by selecting a vectorfnk ∈span{ej |j 1,2, . . . , j /nk}such

thatenk−fnk< dnk1/k, considering4.14and2.6, we arrive at the contradiction:

1enkEAΔnkenk−fnk≤Menk−fnk−→0 ask−→ ∞. 4.17

As follows from the Hahn-Banach Theorem, for anyn1,2, . . ., there is ane∗_n∈X∗such that

e∗_n1, ei, e∗_jδijdi, i, j1,2, . . . . 4.18

Concerning the sequence of the real parts,{Reλn}∞_n1, there are two possibilities: it is either bounded or unbounded above.

Suppose that the sequence{Reλn}∞n1is bounded above, that is, there is such anω >0

that

Reλn≤ω, n1,2, . . . . 4.19

Let

f:

∞

n1

1

n2en. 4.20

By4.14,

EA∪∞_n1Δn

ff, EAΔnf 1

n2en, n1,2, . . . . 4.21

For anyt≥0 and an arbitraryg∗∈X∗,

σAe

tReλ_dv_{f, g}∗_{, λ} _by₄_.₂₁_;

σAe

tReλ_dv_EA_∪∞

n1Δn

f, g∗, λ by2.9;

∞

n1

Δne

tReλ_dv_E

AΔnf, g∗, λ

by4.21;

∞

n1

1

n2

Δne

tReλ_dv_{en, g}∗_{, λ}

forλ∈Δn,by4.12, and4.19,ReλReλn Reλ−Reλn

≤Reλn|λ−λn| ≤ωεn≤ω1;

≤etω1

∞

n1

1

n2v

en, g∗,Δn

by2.8;

≤etω1

∞

n1

1

n24Meng

∗₄_Metω1_g∗∞

n1

1

n2 <∞.

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Similarly, for anyt≥0,

sup

{g∗∈X∗|g∗1}

{λ∈σA|etReλ_>n}

etReλdvf, g∗, λ

≤ sup

{g∗∈X∗|g∗1}

etω1

∞

n1

1

n2

Δn∩{λ∈σA|etReλ_>n}

1dven, g∗, λ

by2.9and4.21;

etω1 sup

{g∗_∈_X∗_|_g∗₁}

1vf, g∗, λ by2.7;

≤etω1 sup

{g∗∈X∗|g∗1}

4MEA

"

λ∈σA|etReλ> n#fg∗

4Metω1EA

"

λ∈σA|etReλ> n#f by the strong continuity of thes.m.;

−→0 asn−→ ∞.

4.23

By2, Proposition3.1,4.22and4.23imply thatf∈₀_≤_t<_∞DetA_.

Thereforesee Section1,yt:etA_f_{, 0}_≤_{t <}_∞_{, is a weak solution of}_1.1_on₀_,_∞_.

Let

h∗:

∞

n1

1

n2e

∗

n∈X∗, 4.24

cf.4.18.

Considering4.18and4.15, we have

en, h∗

dn

n2 ≥

ε

n2, n1,2, . . . . 4.25

Similarly to4.22,

σA|λ|dv

f, h∗, λ

∞

n1

1

n2

Δn|λ|dven, h

∗_{, λ} _for_λ_∈_Δ_n, _by₄_.₁₁_, _|_λ_{| ≥}_n4_;

≥∞

n1

n2ven, h∗,Δn≥

∞

n1

n2|EAΔnen, h∗ | by4.14and 4.25;

≥∞

n1

n2 ε

n2 ∞.

4.26

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Consequently, by Proposition3.1, the weak solutionyt etA_f_{, 0}_≤_{t <}_∞_{, of}_1.1_on

0,∞is not strongly diﬀerentiable at 0.

Now, assume that the sequence{Reλn}∞n1is unbounded above.

Therefore, there is a subsequence{Reλnk}∞k1such that

Reλnk≥k, k1,2, . . . . 4.27

Then the vector

f:

∞

k1

e−nkReλnkenk 4.28

is well defined. By4.14,

EA∪∞_k1Δnkff,

EAΔnkf e−nkReλnkenk, k1,2, . . . .

4.29

For anyt≥0 and an arbitraryg∗∈X∗, similar to4.22,

σAe

tReλ_dv_{f, g}∗_{, λ}∞

k1

e−nkReλnk

Δnke

tReλ_dv_e

nk, g∗, λ

forλ∈Δnk, by4.12, ReλReλnkReλ−Reλnk

≤Reλnkλ−λnk≤Reλnk1;

≤et

∞

k1

e−nk−tReλnkvenk, g∗,Δnk by2.8;

≤et

∞

k1

e−nk−tReλnk4Menkg∗

4Met_g∗∞ k1

e−nk−tReλnk _{by the} _{Comparison Test}_;

<∞.

4.30

Indeed, for all suﬃciently largek’s,nk−t≥1 and by4.27,

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Analogously, for an arbitraryt≥0,

sup

{g∗∈X∗|g∗1}

etReλdvf, g∗, λ≤ sup

{g∗∈X∗|g∗1}

et

∞

k1

e−nk−tReλnk

Δnk∩{λ∈σA|etReλ_>n}

1dvenk, g∗, λet sup

{g∗∈X∗|g∗1}

∞

k1

e−nk/2−tReλnke−nk/2Reλk

1dvenk, g∗, λ

due to4.27,there is an L>0 such thate−nk/2−tReλnk ≤L, k1,2, . . .;

≤Let sup

{g∗_∈_X∗_|_g∗₁}

∞

k1

e−nk/2Reλnk

1dvenk, g∗, λ forh: ∞

k1

e−nk/2Reλnkenk,by4.14and2.9;

Let

1dvh, g∗, λ by2.7;

≤Let sup

{g∗∈|g∗1}

4MEA"λ∈σA|etReλ> n#hg∗

4LMetEA"λ∈σA|etReλ> n#h by the strong continuity of thes.m.;

−→0 asn−→ ∞.

4.32

From4.30and4.32, by2, Proposition3.1, we infer thatf∈₀_≤_t<_∞DetA_.

For anyλ∈Δnk,k1,2, . . ., by4.11,4.12, and4.27,

ReλReλ_nk−Reλ_nk−Reλ

≥Reλnk−Reλnk−Reλ

≥Reλnk−εnk≥Reλnk− 1

nk ≥k−1≥0,

Reλ <max0,2nk−1ln|Imλ|.

4.33

Hence, forλ∈Δnk,k1,2, . . .,

(17)

Using this estimate, we obtain

σA|λ|dv

f, h∗, λ

∞

k1

e−nkReλnk

Δnk|λ|dv

enk, h∗, λ

≥∞

k1

e−nkReλnke2nkReλnk−1/nkvenk, h∗,Δnk

e−2∞ k1

enkReλnk_E

A

Δnkenk, h∗ by4.14, 4.25, and4.27;

≥e−2ε

∞

k1

enk

nk2 ∞.

4.35

Whence, by2, Proposition3.1,y0 f /∈DA.

Therefore, by Proposition3.1, the weak solutionyt etAf, 0 ≤ t < ∞, of1.1on

0,∞is not strongly diﬀerentiable at 0.

With every possibility concerning{Reλn}∞_n1considered, we infer that the opposite to the assumption that, for a certainb >0, the setσA\ Lbis bounded, allows to single out

a weak solution of1.1on0,∞that is not strongly diﬀerentiable at 0, much less strongly infinite diﬀerentiable on0,∞.

Thus, the “only if” part has been proved by contrapositive.

Theorem 5.1 of3has been generalized.

Theorem 4.2. Every weak solution of 1.1on0,∞is strongly infinite diﬀerentiable on0,∞if

and only if there is ab >0 such that, for anyb−>0, the setσA\ Lb−,bis bounded.

Proof. “If” part.

Letb>0 be such that, for an arbitraryb− >0, the setσA\ Lb−,bis bounded.

Lety·be a weak solution of1.1on0,∞. Thensee Section1

yt etAf, 0≤t <∞, 4.36

with somef∈₀_≤_t<_∞DetA_.

For anyn1,2, . . .,t >0, and an arbitraryg∗∈X∗,

σA|λ|

n_etReλ_dv_{f, g}∗_{, λ}

σA\Lb−,b

{λ∈σA∩Lb−,b|Reλ≥0}

{λ∈σA∩Lb−,b|Reλ<0}

|λ|netReλdvf, g∗, λ<∞.

4.37

The first integral in this sum is finite due the boundedness of the setσA\ Lb−,b, the

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The finiteness of the second integral is proved in absolutely the same manner as for the corresponding integral in the proof of the “if” part of Theorem4.1.

Finally for the third one, considering thatb− >0 is arbitrary and settingb−:nt−1, we have

|λ|netReλdvf, g∗, λ

≤

|Reλ||Imλ|netReλdvf, g∗, λ forλ∈σA∩ Lb−,b

with Reλ <0, |Imλ| ≤eb−1−−Reλ_;

≤

−Reλeb−1−−Reλn_etReλ_dv_{f, g}∗_{, λ} _since_x≤_ex_{, x}≥_0;

≤

b−eb−1−−Reλeb−1−−Reλ

n

b−1n

enb−1−−t−Reλ_dv_{f, g}∗_{, λ} _{recall that} _b₋_:_nt−1_;

b−1n

1dvf, g∗, λ≤b−1nvf, g∗, σA by2.8;

≤4Mb−1nfg∗<∞.

4.38

Further, for anyn1,2, . . .andt >0,

sup

{g∗∈X∗|g∗1}

{λ∈σA||λ|n_etReλ_>m}|

λ|netReλdvf, g∗, λ

≤ sup

{g∗∈X∗|g∗1}

{λ∈σA\Lb−,b||λ|netReλ>m}

sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA∩Lb−,b||λ|netReλ>m}

|λ|netReλ_dv_{f, g}∗_{, λ}

−→0 asm−→ ∞.

4.39

Indeed, since, due to the boundedness of the setσA\ Lb−,b, the set{λ ∈σA\ Lb−,b |

|λ|netReλ> m}is void for all suﬃciently largem’s,

sup

{g∗_∈_X∗_|_g∗₁}

{λ∈σA\Lb−,b||λ|netReλ>m}

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Similarly to the “if” part of Theorem4.1and4.38, we have

sup

{g∗∈X∗|g∗1}

{λ∈σA∩Lb−,b||λ|netReλ>m}

|λ|netReλdvf, g∗, λ

≤ sup

{g∗∈X∗|g∗1}

b1n

{λ∈σA∩Lb−,b|Reλ≥0,|λ|netReλ>m}

enb−1tReλ_dv_{f, g}∗_{, λ}

sup

{g∗∈X∗|g∗1}

b−1n

{λ∈σA∩Lb−,b|Reλ<0,|λ|netReλ>m}

enb−1−−t−Reλ_dv_{f, g}∗_{, λ} _with_b₋_:_nt−1_;

sup

{g∗_∈_X∗_|_g∗₁}

b1n

{λ∈σA∩Lb−,b|Reλ≥0,|λ|netReλ>m}

enb−1tReλ_dv_{f, g}∗_{, λ}

sup

{g∗_∈_X∗_|_g∗₁}

b−1n

{λ∈σA∩Lb−,b|Reλ<0,|λ|netReλ>m}

1dvf, g∗, λ

sincef∈

0≤t<∞

D

etA

, by2.7;

≤b1n sup

{g∗_∈_X∗_|_g∗₁}

4MEA

"

λ∈σA∩ Lb−,b |Reλ≥0, |λ|netReλ> m

#

enb−1tA_f_g∗

b−1n sup

{g∗∈X∗|g∗1}

4MEA

"

λ∈σA∩ Lb−,b|Reλ <0, |λ|netReλ> m

#

fg∗

4Mb1nEA

"

λ∈σA∩ Lb−,b|Reλ≥0, |λ|

n_etReλ_{> m}#_enb−1

tA_f

4Mb−1nEA "

λ∈σA∩ Lb−,b|Reλ <0, |λ|netReλ> m

#

f

−→0 as m−→ ∞.

4.41

By the properties of the o.c. and2, Proposition3.1,4.37and4.39imply that

yt etAf∈C∞A, 0< t <∞. 4.42

Whence, by Corollary3.2,y·is strongly infinite diﬀerentiable on0,∞. “Only if” part.

As well as in Theorem4.1, we will prove this part by contrapositive.

Thus, we assume that, for anyb >0, there is such ab−>0 that the setσA\ Lb−,bis

unbounded.

Let us show that this assumption can even be strengthened. To wit: there is such a

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Indeed, there are two possibilities:

1for a certainb−>0, the set{λ∈σA| −b−ln|Imλ|<Reλ≤0}is unbounded;

2for anyb−>0, the set{λ∈σA| −b−ln|Imλ|<Reλ≤0}is bounded.

In the first case, the setσA\ Lb−,bis unbounded with the veryb−>0, for which the set{λ∈σA| −b₋ln|Imλ|<Reλ≤0}is unbounded, and an arbitraryb>0.

In the second case, based on the premise we infer that, for anyb > 0, the set {λ ∈

σA |0 <Reλ < bln|Imλ|}is unbounded. Then so is the setσA\ Lb−,b for anyb− >0

andb>0.

Thus, let us fix ab− > 0 such that the setσA\ Lb−,bis unbounded for an arbitrary

b >0.

In particular, for anyn1,2, . . ., the setσA\ L_b−,2n−1is unbounded.

Hence, we can select a sequence of points in the complex plane{λn}∞n1in the following

manner:

λn∈σA, n1,2, . . . ,

min0,−b−ln|Imλ|<Reλn<max

0,2n−1ln|Imλ| n1,2, . . . ,

λ0:0, |λn|>max

n4,|λn−1|

, n1,2, . . . .

4.43

The latter, in particular, implies that the pointsλnare distinct. Since, for anyn1,2, . . ., the set

"

λ∈ |min0,−b−ln|Imλ|<Reλ <max

0,2n−1ln|Imλ|

,|λ|>max

n4,|λn−1|

#

4.44

is open in , there exists such anεn>0 that this set, along with the pointλn, contains the open

disk of radiusεncentered atλn

Δn:{λ∈ | |λ−λn|< εn}. 4.45

Hence, for anyλ∈Δn,n1,2, . . .,

min0,−b−ln|Imλ|<Reλ <max

0,2n−1ln|Imλ|,

|λ|>maxn4,|λn−1|

.

4.46

Since the pointsλnare distinct, we can regard the radii of the disks,εn, to be small enough so that

0< εn< 1

n, n1,2, . . . , Δi∩Δj ∅, i /j. 4.47

By the properties of the s.m., the latter implies

EAΔiEA

Δj

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Observe that the subspacesEAΔnX,n1,2, . . ., are nontrivial sinceΔn∩σA/∅,Δnbeing

an open set.

By choosing a unit vectoren ∈ EAΔnX en 1,n 1,2, . . ., we obtain a vector

sequence such that

EAΔiejδijei. 4.49

In the same manner as in the proof of Theorem4.1, one can show that there is anε >0 such that

dn:disten, span{ek|k1,2, . . . , k /n}≥ε, n1,2, . . . . 4.50

As follows from the Hahn-Banach Theorem, for anyn1,2, . . ., there is ane∗_n∈X∗such that

e_n∗1, ei, e∗_jδijdi. 4.51

Concerning the sequence of the real parts,{Reλn}∞_n1, there are two possibilities: it is either bounded or unbounded.

First, assume that the sequence{Reλn}∞_n1is bounded, that is, there is anω > 0 such that

|Reλn| ≤ω, n1,2, . . . . 4.52

Let

f:

∞

n1

1

n2en. 4.53

By4.49,

EA

∪∞

n1Δn

ff, EAΔnf

1

n2en, n1,2, . . . . 4.54

In absolutely the same fashion as it was done in the case of bounded above sequence

{Reλn}∞n1in the proof of the “only if” part of Theorem4.1, it is shown thatf ∈0≤t<∞DetA.

Let

h∗:

∞

n1

1

n2e

∗

n∈X∗ 4.55

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Taking4.50and4.51into account, we have

en, h∗ dn

n2 ≥

ε

n2, n1,2, . . . . 4.56

Thus,

σA|λ|e

Reλ_dv_{f, h}∗_{, λ} _considering₄_.₅₄_, _by₂_.₉_;

∞

n1

1

n2

Δn|λ|e

Reλ_dv_{en, h}∗_{, λ} _for_λ_∈_Δ_n, _by₄_.₄₆_, _|_λ_{| ≥}_n4_;

forλ∈Δn, by4.47and 4.52, ReλReλn−Reλn−Reλ

≥Reλn− |Reλn−Reλ| ≥ −ω−εn≥ −ω−1;

≥e−ω1

∞

n1

n2ven, h∗,Δn≥e−ω1

∞

n1

n2|EAΔnen, h∗ | by4.49and 4.56;

≥e−ω1

∞

n1

n2 ε

n2 ∞.

4.57

Whence, by the properties of the o.c. and2, Proposition3.1,y1 eA_{f /}_∈_D_A_.

Consequently, by Proposition3.1, the weak solutionyt etA_f, ₀_≤_{t <}_∞_{, of}_1.1_on

0,∞is not once strongly diﬀerentiable on0,∞.

Now, assume that the sequence {Reλn}∞n1 is unbounded. Therefore, there is a

subsequence{Reλnk}∞_k1such that

Reλnk−→ ∞ or Reλnk −→ −∞ ask−→ ∞. 4.58

Suppose that Reλnk → ∞ask → ∞.

Without restricting generality, we can regard that

Reλnk≥k, k1,2, . . . . 4.59

Considering this case just like the analogous one in the proof of the “only if” part of Theorem4.1, one can show that the vector

f:

∞

k1

e−nkReλnkenk 4.60

belongs to₀_≤_t<_∞DetA_{and, hence,}_{see Section}₁_y_t_:_etA_f_{, 0}_≤_{t <}_∞_{, is a weak solution}

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For anyλ∈Δnk,k1,2, . . ., by4.46,4.47, and4.59,

ReλReλnk−Reλnk−Reλ

≥Reλnk−Reλnk−Reλ

≥Reλnk−εnk≥Reλnk−_n1_k ≥k−1≥0,

Reλ <max0,2nk−1ln|Imλ|,

4.61

|λ| ≥ |Imλ| ≥e2nkReλ≥e2nkReλnk−1/nk. 4.62

Using this estimate, we have

σA|λ|e

Reλ_dv_{f, h}∗_{, λ} _considering₄_.₄₉_, _by₂_.₉_;

∞

k1

e−nkReλnk

Δnk|λ|e

Reλ_dv_enk_{, h}∗_{, λ}

≥∞

k1

e−nkReλnke2nkReλnk−1/nkvenk, h∗,Δnk

e−2

∞

k1

enkReλnkEA

Δnkenk, h∗ by4.49, 4.56, and4.59;

≥e−2ε

∞

k1

enk

nk2 ∞.

4.63

Whence, by the properties of the o.c. and2, Proposition3.1,y1 eA_{f /}_∈_D_A_.

Therefore, by Proposition3.1, the weak solutionyt etAf, 0 ≤ t < ∞, of1.1on

0,∞is not once strongly diﬀerentiable on0,∞. Now, suppose that Reλnk → −∞ask → ∞. Without restricting generality, we can regard that

Reλnk≤ −k, k1,2, . . . . 4.64

Let

f:

∞

k1

1

(24)

For anyt≥0 and an arbitraryg∗∈X∗,

σAe

tReλ_dv_{f, g}∗_{, λ} _considering₄_.₄₉_, _by₂_.₉_;

∞

k1

1

k2

Δnke

tReλ_dv_e

nk, g∗, λ

forλ∈Δnk,by4.42and4.64, ReλReλnkReλ−Reλnk

≤Reλnkλ−λnk≤ −k1≤0;

≤∞

k1

1

k2v

enk, g∗,Δnk by2.8;

≤∞

k1

1

k24Menkg

∗₄_M_g∗∞

k1

1

k2 <∞.

4.66

Analogously, for an arbitraryt≥0,

sup

{g∗∈X∗|g∗1}

≤ sup

{g∗∈X∗|g∗1}

∞

k1

1

k2

1dvenk, g∗, λ

sup

{g∗_∈_X∗_|_g∗₁}

1dvf, g∗, λ by2.7;

≤ sup

{g∗_∈_X∗_|_g∗₁}

4MEA

"

λ∈σA|etReλ> n

#

fg∗

4MEA

"

λ∈σA|etReλ> n#f by the strong continuity of thes.m.;

−→0 asn−→ ∞.

4.67

From4.66and4.67, by2, Proposition3.1, we infer thatf∈₀_≤_t<_∞DetA.

Let

h∗ :

∞

k1

1

k2e

∗

nk∈X∗ 4.68

cf.4.51.

Taking into account4.50and4.51, we have

enk, h∗≥ ε

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For anyλ∈Δnk,k1,2, . . ., by4.47and4.64,

ReλReλnkReλ−Reλnk

≤ReλnkReλ−Reλnk

≤Reλnkεnk≤Reλnk1≤ −k1≤0,

Reλ >min0,−b−ln|Imλ|,

4.70

|λ| ≥ |Imλ| ≥eb−−1−Reλ_. _4.71

Using these estimates, we have

σA|λ|e

b−1

−/2Reλ_dv_{f, h}∗_{, λ} _considering₄_.₄₉_, _by₂_.₉_;

∞

k1

1

k2

Δnk|λ|e

b−1

−/2Reλ_dv_enk_{, h}∗_{, λ}

≥∞

k1

1

k2

Δnke

b−1

−/2−Reλ_dv_e_nk_{, h}∗_{, λ}≥

∞

k1

eb−1−/2k−1

k2 v

enk, h∗,Δnk

≥e−b−1−/2

∞

k1

eb−1−/2k

k2 EA

Δnkenk, h∗ by4.49, 4.64, and4.69;

≥e−b−1−/2_ε

∞

k1

eb−1−/2k

k4 ∞.

4.72

Whence, by the properties of the o.c. and2, Proposition3.1,yb−/2 eb−1−/2A_{f /}∈_D_A_.

Therefore, by Proposition3.1, the weak solutionyt etA_f_{, 0} _≤ _{t <} _∞_{, of}_1.1_on

0,∞is not once strongly diﬀerentiable on0,∞.

With every possibility concerning{Reλn}∞n1considered, we infer that the opposite to

the assumption that there is ab >0 such that, for anyb−>0, the setσA\Lb−,bis bounded,

allows to single out a weak solution of1.1on0,∞that is not once strongly diﬀerentiable on0,∞, much less strongly infinite diﬀerentiable on0,∞.

Thus, the “only if” part has been proved by contrapositive.

Theorem 5.2 of3and Theorem 4.1 of4have been generalized.

5. Certain Effects of Smoothness Improvement

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Therefore, the case of finite strong diﬀerentiability of the weak solutions is superfluous and we obtain the following eﬀects of smoothness improvement.

Proposition 5.1. If every weak solution of 1.1on0,∞is strongly diﬀerentiable at 0, then all of

them are strongly infinite diﬀerentiable on0,∞.

Proposition 5.2. If every weak solution of 1.1on0,∞is once strongly diﬀerentiable on0,∞,

then all of them are strongly infinite diﬀerentiable on0,∞.

These statements generalize Propositions 6.1 and 6.2 of 3, respectively, the latter agreeing with the case whenAis a linear operator not necessarily spectral of scalar type generating aC0-semigroupcf.1,13.

6. Final Remarks

Due to the scalar type spectrality of the operatorA, all the above criteria are formulated exclu-sively in terms of its spectrum, no restrictions on the operator’s resolvent behavior necessary, which makes them inherently qualitative and more transparent than similar results forC0 -semigroupscf.14.

If a scalar type spectralAgenerates aC0-semigroupcf.15, we immediately obtain the results of paper4regarding infinite diﬀerentiability.

Acknowledgment

The author would like to express his gratitude to Mr. Michael and Jonell McCullough.

References

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2401–2422, 2004.

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12 A. Ya. Dorogovtsev, Elements of the General Theory of Measure and Integral, Vyshcha Shkola, Kiev, Ukraine, 1989.

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