On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
Marat V. Markin


On the differentiability of weak solutions]
On the differentiability of weak solutions

of an abstract evolution equation

with a scalar type spectral operator

on the real axis

Marat V. Markin]Marat V. Markin


Department of Mathematics
California State University, Fresno5245 N. Backer Avenue, M/S PB 108Fresno, CA 93740-8001, USA


mmarkin@csufresno.edu









[2010]Primary 34G10, 47B40; Secondary 47B15, 47D06, 47D60


Given the abstract evolution equation

    y'(t)=Ay(t), t∈,

with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on . The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on .


[
    [
    December 30, 2023
=====================


Curiosity is the lust of the mind.Thomas Hobbes



§ INTRODUCTION


We find conditions on a scalar type spectral operator A in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation

    y'(t)=Ay(t), t∈,

which a priori need not be strongly differentiable,
to be strongly infinite differentiable on .
The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case.
We also prove the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on .

The found results develop those of paper <cit.>, where similar consideration is given to the strong differentiability of the weak solutions of the equation

    y'(t)=Ay(t), t≥ 0,

on [0,∞) and (0,∞).

 

Let A be a densely defined closed linear operator in a Banach space X and I be an interval of the real axis . A strongly continuous vector function y:I→ X is called a weak solution of
the evolution equation 

    y'(t)=Ay(t), t∈ I,

if, for any g^* ∈ D(A^*),

    ddt⟨ y(t),g^*⟩ = ⟨ y(t),A^*g^* ⟩, t∈ I,

where D(·) is the domain of an operator, A^* is the operator adjoint to A, and ⟨·,·⟩ is the pairing between
the space X and its dual X^* (cf. <cit.>).




  * Due to the closedness of A, a weak solution of equation (<ref>) can be equivalently defined to be a strongly continuous vector function y:I↦ X such that, for all t∈ I,

    ∫_t_0^ty(s) ds∈ D(A) and y(t)=y(t_0)+A∫_t_0^ty(s) ds,

where t_0 is an arbitrary fixed point of the interval I, and is also called a mild solution (cf. <cit.>, see also <cit.>).


  * Such a notion of weak solution, which need not be differentiable in the strong sense, generalizes that of classical one, strongly differentiable on I and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on I.


  * As is easily seen y:→ X is a weak solution of equation (<ref>) iff 

    y_+(t):=y(t), t≥ 0,

is a weak solution of equation (<ref>) and 

    y_-(t):=y(-t), t≥ 0,

is a weak solution  of the equation

    y'(t)=-Ay(t), t≥ 0.



  * When a closed densely defined linear operator A
in a complex Banach space X generates a strongly continuous group {T(t) }_t∈ of  bounded linear operators (see, e.g., <cit.>), i.e., the associated abstract Cauchy problem (ACP)

    y'(t)=Ay(t), t∈,
    
    y(0)=f

is well-posed (cf. <cit.>), the weak solutions of equation (<ref>) are the orbits

    y(t)=T(t)f, t∈,

with f∈ X (cf. <cit.>, see also <cit.>), whereas the classical ones are those with f∈ D(A)
(see, e.g., <cit.>). 


  * In our discourse, the associated ACP may be ill-posed, i.e., the scalar type spectral operator A need not generate a strongly continuous group of  bounded linear operators (cf. <cit.>). 
 
 




§ PRELIMINARIES


Here, for the reader's convenience, we outline certain essential preliminaries.

Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X,·) with strongly σ-additive spectral measure (the resolution of the identity) E_A(·) assigning to each Borel set δ of the complex plane  a projection operator E_A(δ) on X and having the operator's spectrum σ(A) as its support <cit.>.

Observe that, in a complex finite-dimensional space, 
the scalar type spectral operators are all linear operators on the space, for which there is an eigenbasis (see, e.g., <cit.>) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones <cit.>.

Associated with a scalar type spectral operator in a complex Banach space is the Borel operational calculus analogous to that for a normal operator in a complex Hilbert space <cit.>, which assigns to any Borel measurable function F:σ(A)→ a scalar type spectral operator

    F(A):=∫_σ(A) F(λ) dE_A(λ)

(see <cit.>).

In particular,

    A^n=∫_σ(A)λ^n dE_A(λ), n∈_+,

(_+:={0,1,2,…} is the set of nonnegative integers, A^0:=I, I is the identity operator on X) and

    e^zA:=∫_σ(A) e^zλ dE_A(λ), z∈.


The properties of the spectral measure and operational calculus, exhaustively delineated in <cit.>, underlie the entire subsequent discourse. Here, we underline a few facts of particular importance.

Due to its strong countable additivity, the spectral measure E_A(·) is bounded <cit.>, i.e., there is such an M>0 that, for any Borel set δ⊆,

    E_A(δ)≤ M.


Observe that the notation · is used here to designate the norm in the space L(X) of all bounded linear operators on X. We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space X^*. 


For any f∈ X and g^*∈ X^*, the total variation measure v(f,g^*,·) of the complex-valued Borel measure ⟨ E_A(·)f,g^* ⟩ is a finite positive Borel measure with

    v(f,g^*,)=v(f,g^*,σ(A))≤ 4Mfg^*

(see, e.g., <cit.>).


Also (Ibid.), for a Borel measurable function F:→, f∈ D(F(A)), g^*∈ X^*, and a Borel set δ⊆,

    ∫_δ|F(λ)| dv(f,g^*,λ)
    ≤ 4ME_A(δ)F(A)fg^*.

In particular, for δ=σ(A),
E_A(σ(A))=I and

    ∫_σ(A)|F(λ)| d v(f,g^*,λ)≤ 4MF(A)fg^*.


Observe that the constant M>0 in (<ref>)–(<ref>) is from 
(<ref>).


Further, for a Borel measurable function F:→ [0,∞), a Borel set δ⊆, a sequence {Δ_n}_n=1^∞ 
of pairwise disjoint Borel sets in , and 
f∈ X, g^*∈ X^*,

    ∫_δF(λ) dv(E_A(∪_n=1^∞Δ_n)f,g^*,λ)
    =∑_n=1^∞∫_δ∩Δ_nF(λ) dv(E_A(Δ_n)f,g^*,λ).


Indeed, since, for any Borel sets δ,σ⊆,

    E_A(δ)E_A(σ)=E_A(δ∩σ)

<cit.>, 
for the total variation measure,

    v(E_A(δ)f,g^*,σ)=v(f,g^*,δ∩σ).


Whence, due to the nonnegativity of F(·) (see, e.g., <cit.>),

    ∫_δ F(λ) dv(E_A(∪_n=1^∞Δ_n)f,g^*,λ)
    =∫_δ∩∪_n=1^∞Δ_nF(λ) dv(f,g^*,λ)
    
     =∑_n=1^∞∫_δ∩Δ_nF(λ) dv(f,g^*,λ)
    =∑_n=1^∞∫_δ∩Δ_nF(λ) dv(E_A(Δ_n)f,g^*,λ).


The following statement, allowing to characterize the domains of Borel measurable functions of a scalar type spectral operator in terms of positive Borel measures, is fundamental for our discourse.


 

Let A be a scalar type spectral operator in a complex Banach space (X,·) with spectral measure E_A(·) and F:σ(A)→ be a Borel measurable function. Then f∈ D(F(A)) iff


  (i) for each g^*∈ X^*, 
∫_σ(A) |F(λ)| d v(f,g^*,λ)<∞ and


  (ii) sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | |F(λ)|>n}
|F(λ)| dv(f,g^*,λ)→ 0, n→∞,

where v(f,g^*,·) is the total variation measure of ⟨ E_A(·)f,g^* ⟩.
 

The succeeding key theorem provides a description of the weak solutions of equation (<ref>) with a scalar type spectral operator A in a complex Banach space.

 

Let A be a scalar type spectral operator in a complex Banach space (X,·). A vector function y:[0,∞) → X is a weak solution 
of equation (<ref>) iff there is an f ∈⋂_t≥ 0D(e^tA) such that

    y(t)=e^tAf, t≥ 0,

the operator exponentials understood in the sense of the Borel operational calculus (see (<ref>)).



Theorem <ref> generalizes <cit.>, its counterpart for a normal operator A in a complex Hilbert space.
 

We also need the following characterizations of a particular weak solution's of equation (<ref>) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of [0,∞).

 

Let n∈ and I be a subinterval of [0,∞).  A weak solution y(·) of equation (<ref>) is n times strongly differentiable on I iff

    y(t) ∈ D(A^n), t∈ I,

in which case, 

    y^(k)(t)=A^ky(t), k=1,…,n,t∈ I.




Subsequently, the frequent terms “spectral measure" and “operational calculus" are abbreviated to s.m. and o.c., respectively.




§ GENERAL WEAK SOLUTION


 

Let A be a scalar type spectral operator in a complex Banach space (X,·). A vector function y:→ X is a weak solution 
of equation (<ref>) iff there is an f ∈⋂_t∈D(e^tA) such that

    y(t)=e^tAf, t∈,

the operator exponentials understood in the sense of the Borel operational calculus (see (<ref>)).


  As is noted in the Introduction, y:→ X is a weak solution of (<ref>) iff 

    y_+(t):=y(t), t≥ 0,

is a weak solution of equation (<ref>) and 

    y_-(t):=y(-t), t≥ 0,

is a weak solution of equation (<ref>).

Applying Theorem <ref>, to y_+(·) and y_-(·), we infer that, this is  equivalent to the fact

    y(t)=e^tAf, t∈, with somef ∈⋂_t∈D(e^tA).





  * More generally, Theorem <ref> and its proof can be easily modified to describe in the same manner all weak solution of equation (<ref>) for an arbitrary interval I of the real axis
.


  * Theorem <ref> implies, in particular,


  * that the subspace ⋂_t∈D(e^tA) of all possible initial values of the weak solutions of equation (<ref>) is the largest permissible for the exponential form given by (<ref>), which highlights the naturalness of the notion of weak solution, and


  * that associated ACP (<ref>), whenever solvable,  is solvable uniquely.



  * Observe that the initial-value subspace ⋂_t∈D(e^tA) of equation (<ref>), containing the dense in X subspace ⋃_α>0E_A(Δ_α)X, where

    Δ_α:={λ∈ | |λ|≤α}, α>0,

which coincides with the class ℰ^{0}(A) of entire vectors of A of exponential type <cit.>, is dense in X as well.


  * When a scalar type spectral operator A in a complex Banach space generates a strongly continuous group {T(t) }_t∈ of bounded linear operators, 

    T(t)=e^tA and D(e^tA)=X, t∈,

<cit.>, and hence, Theorem <ref> is consistent with the well-known description of the weak solutions for this setup (see (<ref>)).


  * Clearly, the initial-value subspace ⋂_t∈D(e^tA) of equation (<ref>) is narrower than the initial-value subspace ⋂_t≥ 0D(e^tA) of equation (<ref>) and the initial-value subspace ⋂_t≥ 0D(e^t(-A))=⋂_t≤ 0D(e^tA) of equation (<ref>), in fact it is the intersection of the latter two.
 
 




§ DIFFERENTIABILITY 
OF A PARTICULAR WEAK SOLUTION


Here, we characterize a particular weak solution's of equation (<ref>) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of .

 

Let n∈ and I be a subinterval of . A weak solution y(·) of equation (<ref>) is n times strongly differentiable on I iff

    y(t) ∈ D(A^n), t∈ I,

in which case, 

    y^(k)(t)=A^ky(t), k=1,…,n,t∈ I.



  The statement immediately follows from the prior theorem and Proposition <ref> applied to

    y_+(t):=y(t) and y_-(t):=y(-t), t≥ 0,
 
for an arbitrary weak solution y(·) of equation (<ref>).



Observe that, as well as for Proposition <ref>, for n=1, the subinterval I can degenerate into a singleton.


Inductively, we immediately obtain the following
analog of <cit.>: 

 

Let A be a scalar type spectral operator in a complex Banach space (X,·) and I be a subinterval of . A weak solution y(·) of equation (<ref>) is strongly infinite differentiable on I (y(·)∈ C^∞(I,X)) iff, for each t∈ I, 

    y(t) ∈ C^∞(A):=⋂_n=1^∞D(A^n),

in which case

    y^(n)(t)=A^ny(t), n∈,t∈ I.






§ INFINITE DIFFERENTIABILITY OF WEAK SOLUTIONS


In this section, we characterize the strong infinite differentiability on  of all weak solutions of equation (<ref>) with a scalar type spectral operator A in a complex Banach space.

 

Let A be a scalar type spectral operator in a complex Banach space (X,·) with spectral measure E_A(·). Every weak solution of equation (<ref>) is strongly infinite differentiable on  iff there exist b_+>0 and b_->0 such that the set σ(A)∖ℒ_b_-,b_+,
where

    ℒ_b_-,b_+:={λ∈ | λ≤min(0,-b_-ln|λ|) 
     or λ≥max(0,b_+ln|λ|)},

is bounded 
(see Fig. <ref>).






“If" part.  Suppose that there exist
b_+>0 and b_->0 such that the set σ(A)∖ℒ_b_-,b_+
is bounded and let y(·) be an arbitrary weak solution of equation (<ref>). 

By Theorem <ref>, 

    y(t)=e^tAf, t∈, with somef ∈⋂_t∈D(e^tA).


Our purpose is to show that y(·)∈ C^∞(,X), which, by Corollary <ref>, is attained by showing that, for each t∈,

    y(t)∈ C^∞(A):=⋂_n=1^∞ D(A^n).


Let us proceed by proving that, for any t∈ and m∈

    y(t)∈ D(A^m)
 
via Proposition <ref>.

For any t∈, m∈ and an arbitrary g^*∈ X^*,

    ∫_σ(A)|λ|^me^tλ dv(f,g^*,λ)
    =∫_σ(A)∖ℒ_b_-,b_+|λ|^me^tλ dv(f,g^*,λ)
    
    
    +∫_{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1 }|λ|^me^tλ dv(f,g^*,λ)
    
    
    +∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }|λ|^me^tλ dv(f,g^*,λ)
    
    
    +∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }|λ|^me^tλ dv(f,g^*,λ)<∞.


Indeed, 

    ∫_σ(A)∖ℒ_b_-,b_+|λ|^me^tλ dv(f,g^*,λ)<∞

and

    ∫_{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1 }|λ|^me^tλ dv(f,g^*,λ)<∞

due to the boundedness of the sets

    σ(A)∖ℒ_b_-,b_+ and{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1 },

the continuity of the integrated function on , and the finiteness of the measure v(f,g^*,·).

Further, for any t∈, m∈ and an arbitrary g^*∈ X^*,

    ∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }|λ|^me^tλ dv(f,g^*,λ)
    
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }[|λ|+|λ|]^me^tλ dv(f,g^*,λ)
    
    since, for λ∈σ(A)∩ℒ_b_-,b_+ with λ≥ 1, e^b_+^-1λ≥ |λ|;
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }[λ+e^b_+^-1λ]^me^tλ dv(f,g^*,λ)
    
    since, in view of λ≥ 1, b_+e^b_+^-1λ≥λ;
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }[b_+e^b_+^-1λ+e^b_+^-1λ]^me^tλ dv(f,g^*,λ)
    
    
    = [b_++1]^m∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1 }e^[mb_+^-1+t]λ dv(f,g^*,λ)
    
    since f∈⋂_t∈D(e^tA), by Proposition <ref>;
    
    <∞.


Finally, for any t∈, m∈ and an arbitrary g^*∈ X^*,

    ∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }|λ|^me^tλ dv(f,g^*,λ)
    
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }[|λ|+|λ|]^me^tλ dv(f,g^*,λ)
    
    since, for λ∈σ(A)∩ℒ_b_-,b_+ with λ≤ -1, e^b_-^-1(-λ)≥ |λ|;
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }[-λ+e^b_-^-1(-λ)]^me^tλ dv(f,g^*,λ)
    
    since, in view of -λ≥ 1, b_-e^b_-^-1(-λ)≥-λ;
    ≤∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }[b_-e^b_-^-1(-λ)+e^b_-^-1(-λ)]^me^tλ dv(f,g^*,λ)
    
    
    = [b_-+1]^m∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1 }e^[t-mb_-^-1]λ dv(f,g^*,λ)
    
    since f∈⋂_t∈D(e^tA), by Proposition <ref>;
    
    <∞.


Also, for any t∈, m∈ and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | |λ|^me^tλ>n}
    |λ|^me^tλ dv(f,g^*,λ)
    
    ≤sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)
    ∖ℒ_b_-,b_+ | |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    
    + sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    
    + sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    
    + sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    → 0, n→∞.


Indeed, since, due to the boundedness of the sets

    σ(A)∖ℒ_b_-,b_+ and{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1}

and the continuity of the integrated function on ,
the sets

    {λ∈σ(A)∖ℒ_b_-,b_+ | |λ|^me^tλ>n}

and 

    {λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1,  |λ|^me^tλ>n}

are empty for all sufficiently large n∈,
we immediately infer that, for any t∈ and m∈,

    lim_n→∞sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)
    ∖ℒ_b_-,b_+ | |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)=0

and

    lim_n→∞sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | -1<λ<1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    =0.


Further, for any t∈, m∈ and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    as in (<ref>);
    ≤sup_{g^*∈ X^* | g^*=1}
    [b_++1]^m∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1,  |λ|^me^tλ>n}e^[mb_+^-1+t]λ dv(f,g^*,λ)
    
    since f∈⋂_t∈D(e^tA), by (<ref>);
    ≤sup_{g^*∈ X^* | g^*=1}
    
    [b_++1]^m4ME_A({λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1,  |λ|^me^tλ>n})
    e^[mb_+^-1+t]Afg^*
    ≤ [b_++1]^m4ME_A({λ∈σ(A)∩ℒ_b_-,b_+ | λ≥ 1,  |λ|^me^tλ>n})
    e^[mb_+^-1+t]Af
    by the strong continuity of the  s.m.;
     → [b_++1]^m4ME_A(∅)e^[mb_+^-1+t]Af=0, n→∞.


Finally, for any t∈, m∈ and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1,  |λ|^me^tλ>n}|λ|^me^tλ dv(f,g^*,λ)
    
    as in (<ref>);
    ≤sup_{g^*∈ X^* | g^*=1}
    [b_-+1]^m∫_{λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1,  |λ|^me^tλ>n}e^[t-mb_-^-1]λ dv(f,g^*,λ)
    
    since f∈⋂_t∈D(e^tA), by (<ref>);
    ≤sup_{g^*∈ X^* | g^*=1}
    
    [b_-+1]^m4ME_A({λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1,  |λ|^me^tλ>n})
    e^[t-mb_-^-1]Afg^*
    ≤ [b_-+1]^m4ME_A({λ∈σ(A)∩ℒ_b_-,b_+ | λ≤ -1,  |λ|^me^tλ>n})
    e^[t-mb_-^-1]Af
    by the strong continuity of the  s.m.;
     → [b_++1]^m4ME_A(∅)e^[t-mb_-^-1]Af=0, n→∞.


By Proposition <ref> and the properties of the o.c. (see <cit.>), (<ref>) and (<ref>) jointly imply that, for any t∈ and m∈,

    f∈ D(A^me^tA),

which further implies that, for each t∈, 

    y(t)=e^tAf∈⋂_n=1^∞ D(A^n)
    =:C^∞(A).


Whence, by Corollary <ref>, we infer that

    y(·) ∈ C^∞(,X),

which completes the proof of the “if" part.


“Only if" part.  Let us prove this part by contrapositive assuming that, for any b_+>0 and b_->0, the set 
σ(A)∖ℒ_b_-,b_+ is unbounded. In particular, this means that, for any n∈, unbounded is the set

    σ(A)∖ℒ_(2n)^-1,(2n)^-1=
    {λ∈σ(A) | 
    -(2n)^-1ln|λ|<λ < (2n)^-1ln|λ|}.
 

Hence, we can choose a sequence of points {λ_n}_n=1^∞ 
in the complex plane as follows:

    λ_n ∈σ(A), n∈,
       -(2n)^-1ln|λ_n|<λ_n < (2n)^-1ln|λ_n|, n∈,
       λ_0:=0, |λ_n|>max[n^4,|λ_n-1|], n∈.


The latter implies, in particular, that the points λ_n, n∈, are distinct (λ_i ≠λ_j, i≠ j).

Since, for each n∈, the set

    {λ∈ℂ |  
    -(2n)^-1ln|λ|<λ < (2n)^-1ln|λ|,|λ|>max[n^4,|λ_n-1|]}

is open in , along with the point λ_n, it contains an open disk

    Δ_n:={λ∈ | |λ-λ_n|<ε_n }
 
centered at λ_n of some radius ε_n>0, i.e., for each λ∈Δ_n,

    -(2n)^-1ln|λ|<λ < (2n)^-1ln|λ| and |λ|>max[n^4,|λ_n-1|].


Furthermore, we can regard the radii of the disks to be small enough so that

    0<ε_n<1n, n∈, and
       Δ_i ∩Δ_j=∅, i≠ j
      (i.e., the disks are  pairwise disjoint).


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

    E_A(Δ_i)E_A(Δ_j)=0, i≠ j,

where 0 stands for the zero operator on X.

Observe also, that the subspaces E_A(Δ_n)X, n∈, are nontrivial since

    Δ_n ∩σ(A)≠∅, n∈,

with Δ_n being an open set in . 

By choosing a unit vector e_n∈ E_A(Δ_n)X for each n∈, we obtain a sequence 
{e_n}_n=1^∞ in X such that

    e_n=1, n∈, and E_A(Δ_i)e_j=δ_ije_j, i,j∈,

where δ_ij is the Kronecker delta.

As is easily seen, (<ref>) implies that the vectors e_n, n∈, are linearly independent.

Furthermore, there is an ε>0 such that

    d_n:=(e_n,({e_i | i∈, i≠ n}))≥ε, n∈.


Indeed, the opposite implies the existence of a subsequence {d_n(k)}_k=1^∞ such that

    d_n(k)→ 0, k→∞.


Then, by selecting a vector

    f_n(k)∈({e_i | i∈, i≠ n(k)}), k∈,
 
such that 

    e_n(k)-f_n(k)<d_n(k)+1/k, k∈,

we arrive at

    1=e_n(k)since, by (<ref>), 
    E_A(Δ_n(k))f_n(k)=0;
    
    =E_A(Δ_n(k))(e_n(k)-f_n(k))≤E_A(Δ_n(k))e_n(k)-f_n(k)by (<ref>);
     ≤ Me_n(k)-f_n(k)≤ M[d_n(k)+1/k]
    → 0, k→∞,

which is a contradiction proving (<ref>). 

As follows from the Hahn-Banach Theorem, for any n∈, there is an e^*_n∈ X^* such that 

    e_n^*=1, n∈, and ⟨ e_i,e_j^*⟩=δ_ijd_i, i,j∈.



Let us consider separately the two possibilities concerning the sequence of the real parts {λ_n}_n=1^∞: its being bounded or unbounded. 

First, suppose that the sequence {λ_n}_n=1^∞ is bounded, i.e., there is such an ω>0 that

    |λ_n| ≤ω, n∈,

and consider the element

    f:=∑_k=1^∞ k^-2e_k∈ X,

which is well defined since {k^-2}_k=1^∞∈ l_1 (l_1 is the space of absolutely summable sequences) and e_k=1, k∈ (see (<ref>)).

In view of (<ref>), by the properties of the s.m.,

    E_A(∪_k=1^∞Δ_k)f=f and E_A(Δ_k)f=k^-2e_k, k∈.


For any t≥ 0 and an arbitrary g^*∈ X^*,

    ∫_σ(A)e^tλ dv(f,g^*,λ)
    by (<ref>);
    
    =∫_σ(A) e^tλ d v(E_A(∪_k=1^∞Δ_k)f,g^*,λ)
    by (<ref>);
    
    =∑_k=1^∞∫_σ(A)∩Δ_ke^tλ dv(E_A(Δ_k)f,g^*,λ)
    by (<ref>);
    
    =∑_k=1^∞ k^-2∫_σ(A)∩Δ_ke^tλ dv(e_k,g^*,λ)
    
    since, for λ∈Δ_k, by (<ref>) and (<ref>), λ=λ_k+(λ-λ_k)
    
    ≤λ_k+|λ-λ_k|≤ω+ε_k≤ω+1;
    
    ≤ e^t(ω+1)∑_k=1^∞ k^-2∫_σ(A)∩Δ_k1 dv(e_k,g^*,λ)
    = e^t(ω+1)∑_k=1^∞ k^-2v(e_k,g^*,Δ_k)
    
    by (<ref>);
    ≤ e^t(ω+1)∑_k=1^∞ k^-24Me_kg^*
    = 4Me^t(ω+1)g^*∑_k=1^∞ k^-2<∞.
 

Also, for any t<0 and an arbitrary g^*∈ X^*,

    ∫_σ(A)e^tλ dv(f,g^*,λ)
    by (<ref>);
    
    =∫_σ(A) e^tλ d v(E_A(∪_k=1^∞Δ_k)f,g^*,λ)
    by (<ref>);
    
    =∑_k=1^∞∫_σ(A)∩Δ_ke^tλ dv(E_A(Δ_k)f,g^*,λ)
    by (<ref>);
    
    =∑_k=1^∞ k^-2∫_σ(A)∩Δ_ke^tλ dv(e_k,g^*,λ)
    
    since, for λ∈Δ_k, by (<ref>) and (<ref>), λ=λ_k-(λ_k-λ)
    
    ≥λ_k-|λ_k-λ|≥ -ω-ε_k≥ -ω-1;
    
    ≤ e^-t(ω+1)∑_k=1^∞ k^-2∫_σ(A)∩Δ_k1 dv(e_k,g^*,λ)
    = e^-t(ω+1)∑_k=1^∞ k^-2v(e_k,g^*,Δ_k)
    
    by (<ref>);
    ≤ e^-t(ω+1)∑_k=1^∞ k^-24Me_kg^*
    = 4Me^-t(ω+1)g^*∑_k=1^∞ k^-2<∞.
 

Similarly, to (<ref>) for any t≥ 0 and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n} 
    e^tλ dv(f,g^*,λ)
    
    ≤sup_{g^*∈ X^* | g^*=1}e^t(ω+1)∑_k=1^∞ k^-2∫_{λ∈σ(A) | e^tλ>n}∩Δ_k1 dv(e_k,g^*,λ) 
    
    by (<ref>);
    
    =e^t(ω+1)sup_{g^*∈ X^* | g^*=1}∑_k=1^∞∫_{λ∈σ(A) | e^tλ>n}∩Δ_k1 dv(E_A(Δ_k)f,g^*,λ) 
    
    by (<ref>);
    
    = e^t(ω+1)sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(E_A(∪_k=1^∞Δ_k)f,g^*,λ)
    
    by (<ref>);
    
    = e^t(ω+1)sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(f,g^*,λ)
    by (<ref>);
    ≤ e^t(ω+1)sup_{g^*∈ X^* | g^*=1}4ME_A({λ∈σ(A) | e^tλ>n})fg^*
    ≤ 4Me^t(ω+1)E_A({λ∈σ(A) | e^tλ>n})f
    by the strong continuity of the  s.m.;
    → 4Me^t(ω+1)E_A(∅)f=0, n→∞.


Similarly, to (<ref>) for any t<0 and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n} 
    e^tλ dv(f,g^*,λ)
    
    ≤sup_{g^*∈ X^* | g^*=1}e^-t(ω+1)∑_k=1^∞ k^-2∫_{λ∈σ(A) | e^tλ>n}∩Δ_k1 dv(e_k,g^*,λ) 
    
    by (<ref>);
    
    =e^-t(ω+1)sup_{g^*∈ X^* | g^*=1}∑_k=1^∞∫_{λ∈σ(A) | e^tλ>n}∩Δ_k1 dv(E_A(Δ_k)f,g^*,λ) 
    
    by (<ref>);
    
    = e^-t(ω+1)sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(E_A(∪_k=1^∞Δ_k)f,g^*,λ)
    
    by (<ref>);
    
    = e^-t(ω+1)sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(f,g^*,λ)
    by (<ref>);
    ≤ e^-t(ω+1)sup_{g^*∈ X^* | g^*=1}4ME_A({λ∈σ(A) | e^tλ>n})fg^*
    ≤ 4Me^-t(ω+1)E_A({λ∈σ(A) | e^tλ>n})f
    by the strong continuity of the  s.m.;
    → 4Me^-t(ω+1)E_A(∅)f=0, n→∞.


By Proposition <ref>, (<ref>), (<ref>), (<ref>), and (<ref>) jointly imply that 

    f∈⋂_t∈D(e^tA),

and hence, by Theorem <ref>,

    y(t):=e^tAf, t∈,

is a weak solution of equation (<ref>).

Let

    h^*:=∑_k=1^∞ k^-2e_k^*∈ X^*,

the functional being well defined since {k^-2}_k=1^∞∈ l_1 and e_k^*=1, k∈ (see (<ref>)).

In view of (<ref>) and (<ref>), we have:

    ⟨ e_n,h^*⟩=⟨ e_k,k^-2e_k^*⟩=d_k k^-2≥ε k^-2, k∈.


Hence,

    ∫_σ(A)|λ| dv(f,h^*,λ)
    by (<ref>) as in (<ref>);
    
    =∑_k=1^∞ k^-2∫_σ(A)∩Δ_k|λ| dv(e_k,h^*,λ)
    
    since, for λ∈Δ_k, by
    (<ref>), |λ|≥ k^4;
    ≥∑_k=1^∞ k^-2k^4 v(e_k,h^*,Δ_k)≥∑_k=1^∞ k^2|⟨ E_A(Δ_k)e_k,h^*⟩|
    
    by (<ref>) and (<ref>);
    ≥∑_k=1^∞ k^2 ε k^-2=∞.
 

By Proposition <ref>, (<ref>) implies that

    y(0)=f∉ D(A),

which, by Proposition <ref> (n=1, I={0}) further implies that the weak solution y(t)=e^tAf, t∈, of equation (<ref>) is not strongly differentiable at 0. 


Now, suppose that the sequence {λ_n}_n=1^∞
is unbounded. 

Therefore, there is a subsequence {λ_n(k)}_k=1^∞ such that

    λ_n(k)→∞ or λ_n(k)→ -∞, k→∞.

Let us consider separately each of the two cases.

First, suppose that 

    λ_n(k)→∞, k→∞
 
Then, without loss of generality, we can regard that

    λ_n(k)≥ k, k∈.


Consider the elements

    f:=∑_k=1^∞ e^-n(k)λ_n(k)e_n(k)∈ X
     and h:=∑_k=1^∞ e^-n(k)/2λ_n(k)e_n(k)∈ X,

well defined since, by (<ref>),

    {e^-n(k)λ_n(k)}_k=1^∞,
    {e^-n(k)/2λ_n(k)}_k=1^∞∈ l_1

and e_n(k)=1, k∈ (see (<ref>)).

By (<ref>),

    E_A(∪_k=1^∞Δ_n(k))f=f andE_A(Δ_n(k))f=e^-n(k)λ_n(k)e_n(k),k∈,

and

    E_A(∪_k=1^∞Δ_n(k))h=h andE_A(Δ_n(k))h=e^-n(k)/2λ_n(k)e_n(k), k∈.


For any t≥ 0 and an arbitrary g^*∈ X^*, 

    ∫_σ(A)e^tλ dv(f,g^*,λ)
    by (<ref>) as in (<ref>);
    
    =∑_k=1^∞ e^-n(k)λ_n(k)∫_σ(A)∩Δ_n(k)e^tλ dv(e_n(k),g^*,λ)
    
    since, for λ∈Δ_n(k), by (<ref>), λ
    =λ_n(k)+(λ-λ_n(k))
    
    ≤λ_n(k)+|λ-λ_n(k)|≤λ_n(k)+1;
    
    ≤∑_k=1^∞ e^-n(k)λ_n(k)
    e^t(λ_n(k)+1)∫_σ(A)∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    
    = e^t∑_k=1^∞ e^-[n(k)-t]λ_n(k)v(e_n(k),g^*,Δ_n(k))
    by (<ref>);
    ≤ e^t∑_k=1^∞ e^-[n(k)-t]λ_n(k)4Me_n(k)g^*
    = 4Me^tg^*∑_k=1^∞ e^-[n(k)-t]λ_n(k)
    
    <∞.


Indeed, for all k∈ sufficiently large so that

    n(k)≥ t+1,

in view of (<ref>), 

    e^-[n(k)-t]λ_n(k)≤ e^-k.


For any t<0 and an arbitrary g^*∈ X^*, 

    ∫_σ(A)e^tλ dv(f,g^*,λ)
    by (<ref>) as in (<ref>);
    
    =∑_k=1^∞ e^-n(k)λ_n(k)∫_σ(A)∩Δ_n(k)e^tλ dv(e_n(k),g^*,λ)
    
    since, for λ∈Δ_n(k), by (<ref>), λ
    =λ_n(k)-(λ_n(k)-λ)
    
    ≥λ_n(k)-|λ_n(k)-λ|≥λ_n(k)-1;
    
    ≤∑_k=1^∞ e^-n(k)λ_n(k)
    e^t(λ_n(k)-1)∫_σ(A)∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    
    = e^-t∑_k=1^∞ e^-[n(k)-t]λ_n(k)v(e_n(k),g^*,Δ_n(k))
    by (<ref>);
    ≤ e^-t∑_k=1^∞ e^-[n(k)-t]λ_n(k)4Me_n(k)g^*
    = 4Me^-tg^*∑_k=1^∞ e^-[n(k)-t]λ_n(k)
    
    <∞.


Indeed, for all k∈, in view of t<0,

    n(k)-t≥ n(k)≥ 1,

and hence, in view of (<ref>), 

    e^-[n(k)-t]λ_n(k)≤ e^-k.


Similarly to (<ref>), for any t≥ 0 and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}e^tλ dv(f,g^*,λ)
    
    ≤sup_{g^*∈ X^* | g^*=1}e^t∑_k=1^∞ e^-[n(k)-t]λ_n(k)∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    
    =e^tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞ e^-[n(k)/2-t]λ_n(k)
    e^-n(k)/2λ_(k)
    ∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    since, by (<ref>), there is an L>0 such that
    e^-[n(k)/2-t]λ_n(k)≤ L, k∈;
    ≤ Le^tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞ e^-n(k)/2λ_n(k)∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    by (<ref>);
    
    = Le^tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(E_A(Δ_n(k))h,g^*,λ)
    
    by (<ref>);
    
    = Le^tsup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(E_A(∪_k=1^∞Δ_n(k))h,g^*,λ)
    
    by (<ref>);
    
    =Le^tsup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(h,g^*,λ)
    by (<ref>);
    ≤ Le^tsup_{g^*∈ X^* | g^*=1}4M
    E_A({λ∈σ(A) | e^tλ>n})hg^*
    ≤ 4LMe^tE_A({λ∈σ(A) | e^tλ>n})h
    by the strong continuity of the  s.m.;
    → 4LMe^tE_A(∅)h=0, n→∞.


Similarly to (<ref>), for any t<0 and an arbitrary n∈,

    sup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}e^tλ dv(f,g^*,λ)
    
    ≤sup_{g^*∈ X^* | g^*=1}e^-t∑_k=1^∞ e^-[n(k)-t]λ_n(k)∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    
    =e^-tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞ e^-[n(k)/2-t]λ_n(k)
    e^-n(k)/2λ_(k)
    ∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    since, by (<ref>), there is an L>0 such that
    e^-[n(k)/2-t]λ_n(k)≤ L, k∈;
    ≤ Le^-tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞ e^-n(k)/2λ_n(k)∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(e_n(k),g^*,λ)
    
    by (<ref>);
    
    = Le^-tsup_{g^*∈ X^* | g^*=1}∑_k=1^∞∫_{λ∈σ(A) | e^tλ>n}∩Δ_n(k)1 dv(E_A(Δ_n(k))h,g^*,λ)
    
    by (<ref>);
    
    = Le^-tsup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(E_A(∪_k=1^∞Δ_n(k))h,g^*,λ)
    
    by (<ref>);
    
    =Le^-tsup_{g^*∈ X^* | g^*=1}∫_{λ∈σ(A) | e^tλ>n}1 dv(h,g^*,λ)
    by (<ref>);
    ≤ Le^-tsup_{g^*∈ X^* | g^*=1}4M
    E_A({λ∈σ(A) | e^tλ>n})hg^*
    ≤ 4LMe^-tE_A({λ∈σ(A) | e^tλ>n})h
    by the strong continuity of the  s.m.;
    → 4LMe^-tE_A(∅)h=0, n→∞.


By Proposition <ref>, (<ref>), (<ref>), (<ref>), and (<ref>) jointly imply that 

    f∈⋂_t∈D(e^tA),

and hence, by Theorem <ref>,

    y(t):=e^tAf, t∈,

is a weak solution of equation (<ref>).

Since, for any λ∈Δ_n(k), k∈, by (<ref>), (<ref>),

    λ =λ_n(k)-(λ_n(k)-λ)
    ≥λ_n(k)-|λ_n(k)-λ|
    
      ≥λ_n(k)-ε_n(k)≥λ_n(k)-1/n(k)≥ k-1≥ 0

and, by (<ref>),

    λ<(2n(k))^-1ln|λ|,

we infer that, for any λ∈Δ_n(k), k∈,

    |λ|≥|λ|≥ 
    e^2n(k)λ≥ e^2n(k)(λ_n(k)-1/n(k)).


Using this estimate, for the functional h^*∈ X^* defined by (<ref>), we have:

    ∫_σ(A)|λ| dv(f,h^*,λ)
    by (<ref>) as in (<ref>);
    
    =∑_k=1^∞ e^-n(k)λ_n(k)∫_Δ_n(k)|λ| dv(e_n(k),h^*,λ)
    
    ≥∑_k=1^∞ e^-n(k)λ_n(k)e^2n(k)(λ_n(k)-1/n(k))v(e_n(k),h^*,Δ_n(k))
    
    
    = ∑_k=1^∞
    e^-2 e^n(k)λ_n(k)|⟨ E_A(Δ_n(k))e_n(k),h^*⟩|
    by (<ref>), (<ref>), and (<ref>);
    ≥∑_k=1^∞ e^-2εe^n(k)n(k)^2=∞.
 

By Proposition <ref>, (<ref>) implies that

    y(0)=f∉ D(A),

which, by Proposition <ref> (n=1, I={0}), further implies that the weak solution y(t)=e^tAf, t∈, 
of equation (<ref>) is not strongly differentiable at 0. 


The remaining case of

    λ_n(k)→ -∞, k→∞
 
is symmetric to the case of

    λ_n(k)→∞, k→∞
 
and is considered in absolutely the same manner, which furnishes a weak solution y(·) of equation (<ref>) such that

    y(0)∉D(A),

and hence, by Proposition <ref> (n=1, I={0}), not strongly differentiable at 0.

With every possibility concerning {λ_n}_n=1^∞ considered, 
we infer that assuming the opposite to the “if" part's premise allows to find a weak solution of (<ref>) on [0,∞)
that is not strongly differentiable at 0, and hence, much less strongly infinite differentiable on .

Thus, the proof by contrapositive of the “only if" part is complete and so is the proof of the 
entire statement


From Theorem <ref> and <cit.>, the latter characterizing the strong infinite differentiability of all weak solution
of equation (<ref>) on (0,∞), we also obtain 


Let A be a scalar type spectral operator in a complex Banach space. If all weak solutions of equation (<ref>) are strongly infinite differentiable on (0,∞), then all weak solutions of equation (<ref>) are strongly infinite differentiable on .



As follows from Theorem <ref>, all weak solutions of equation (<ref>) with a scalar type spectral operator A in a complex Banach space can 
be strongly infinite differentiable while the operator A is unbounded, e.g., when A is an unbounded self-adjoint operator in a complex Hilbert space (cf. <cit.>). This fact contrasts the situation when a closed densely defined linear operator A in a complex Banach space generates a strongly continuous group {T(t) }_t∈ of bounded linear operators, i.e., the associated abstract Cauchy problem is well-posed (see Remarks <ref>), in which case even the (left or right) strong differentiability of all weak solutions of equation (<ref>) at 0 immediately implies boundedness for A (cf. <cit.>).





§ THE CASES OF NORMAL AND SELF-ADJOINT OPERATORS


As an important particular case of Theorem <ref>, we obtain

 

Let A be a normal operator in a complex Hilbert space. Every weak solution of equation (<ref>) is strongly infinite differentiable on  iff there exist 
b_+>0 and b_->0 such that the set σ(A)∖ℒ_b_-,b_+,
where

    ℒ_b_-,b_+:={λ∈ | λ≤min(0,-b_-ln|λ|) 
     or λ≥max(0,b_+ln|λ|)},

is bounded (see Fig. <ref>).



Corollary <ref> develops the results of paper <cit.>, where similar consideration is given to the strong differentiability of the weak solutions of equation (<ref>) with a normal operator A in a complex Hilbert space on [0,∞) and (0,∞).


From Corollary <ref>, we immediately obtain the following


Let A be a normal operator in a complex Hilbert space. If all weak solutions of equation (<ref>) are strongly infinite differentiable on (0,∞) (cf. <cit.>), then all weak solutions of equation (<ref>) are strongly infinite differentiable on .


Considering that, for a self-adjoint operator A in a complex Hilbert space X,

    σ(A)⊆

(see, e.g., <cit.>), we further arrive at

 

Every weak solution of equation (<ref>) 
with a self-adjoint operator A in a complex Hilbert space is strongly infinite differentiable on .


Cf. <cit.>.




§ INHERENT SMOOTHNESS IMPROVEMENT EFFECT


As is observed in the proof of the “only if" part of Theorem <ref>, the opposite to the “if" part's premise implies that there is a weak solution of equation (<ref>), which is not strongly differentiable at 0. This renders the case of finite strong  differentiability of the weak solutions superfluous and we arrive at the following inherent effect of smoothness improvement.


Let A be a scalar type spectral operator in a complex Banach space (X,·). If every weak solution of equation (<ref>) is strongly differentiable at 0, then all of them are strongly infinite differentiable on .


Cf. <cit.>.




§ CONCLUDING REMARK


Due to the scalar type spectrality of the operator A, Theorem <ref> is stated exclusively in terms of the location of its spectrum in the complex plane, similarly to the celebrated Lyapunov stability theorem <cit.> 
(cf. <cit.>),
and thus, is an intrinsically qualitative statement (cf. <cit.>).




§ ACKNOWLEDGMENTS


The author extends sincere appreciation to his colleague, Dr. Maria Nogin of the Department of Mathematics, California State University, Fresno, for her kind assistance with the graphics.




§ CONFLICTS OF INTEREST


The author declares that there are no conflicts of interest regarding the publication of this paper.


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no. 3-4,	
573–606.	

Markin2002(1)
,	
On an abstract evolution equation with a spectral operator of scalar type,	
Int. J. Math. Math. Sci.	
32	
(2002),	
no. 9,	
555–563.	

Markin2002(2)
,	
A note on the spectral operators of scalar type and semigroups of bounded linear operators,	
Ibid.	
32	
(2002),	
no. 10,	
635–640.	

Markin2004(1)
,	
On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiable C_0-semigroups,	
Ibid.	
2004	
(2004),	
no. 45,	
2401–2422.	

Markin2004(2)
,	
On the Carleman classes of vectors of a scalar type spectral operator,	
Ibid.	
2004	
(2004),	
no. 60,	
3219–3235.	

Markin2011
,	
On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator,	
Ibid.	
2011	
(2011),	
Article ID 825951,	
27 pp.	


Markin2015
,	
On the Carleman ultradifferentiable vectors of a scalar type spectral operator,	
Methods Funct. Anal. Topology 
21	
(2015),	
no. 4,	
361–369.	

Markin2018(2)
,	
On the mean ergodicity of weak solutions of an abstract evolution equation,	
Ibid.	
24	
(2018),	
no. 1,	
53–70.	


Pazy1968
A. Pazy,	
On the differentiability and compactness of semigroups of linear operators,	
J. Math. Mech.	
17	
(1968),	
no. 12,	
1131–1141.	

Plesner
A.I. Plesner,	
Spectral Theory of Linear Operators,	

Nauka,	
Moscow,		
1965		
(Russian).

Wermer
J. Wermer,	
Commuting spectral measures on Hilbert space,	
Pacific J. Math.	
4	
(1954),	
no. 3,	
355–361.	



