International Journal of Mathematics and Mathematical Sciences

Volume 2018, Article ID 4168609, 14 pages

https://doi.org/10.1155/2018/4168609

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

Department of Mathematics, California State University, Fresno, 5245 N. Backer Avenue, M/S PB 108, Fresno, CA 93740-8001, USA

Correspondence should be addressed to Marat V. Markin; ude.onserfusc@nikramm

Received 27 March 2018; Revised 10 May 2018; Accepted 20 June 2018; Published 17 July 2018

Academic Editor: Seppo Hassi

Copyright © 2018 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.

#### Abstract

Given the abstract evolution equation , with* scalar type spectral operator * 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 * 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 , then all of them are strongly infinite differentiable on .

*“Curiosity is the lust of the mind.”*

Thomas Hobbes

#### 1. Introduction

We find conditions on a* scalar type spectral* operator in a complex Banach space necessary and sufficient for all* weak solutions* of the evolution equationwhich a priori need not be strongly differentiable, to be strongly infinite differentiable on . The important case of the equation with a* normal operator* 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 [1], where similar consideration is given to the strong differentiability of the weak solutions of the equationon and .

*Definition 1 (weak solution). *Let be a densely defined closed linear operator in a Banach space and be an interval of the real axis . A strongly continuous vector function is called a* weak solution* of the evolution equationif, for any , where is the* domain* of an operator, is the operator* adjoint* to , and is the* pairing* between the space and its dual (cf. [2]).

*Remarks 2. *(i)Due to the* closedness* of , a weak solution of (3) can be equivalently defined to be a strongly continuous vector function such that, for all , where is an arbitrary fixed point of the interval , and is also called a* mild solution* (cf. [3, Ch. II, Definition 6.3], see also [4, Preliminaries]).(ii)Such a notion of* weak solution*, which need not be differentiable in the strong sense, generalizes that of* classical* one, strongly differentiable on and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on .(iii)As is easily seen is a weak solution of (1)* iff* is a weak solution of (2) and is a weak solution of the equation(iv)When a closed densely defined linear operator in a complex Banach space generates a strongly continuous group of bounded linear operators (see, e.g., [3, 5]), i.e., the associated* abstract Cauchy problem* (*ACP*) is* well-posed* (cf. [3, Ch. II, Definition 6.8]), the weak solutions of (1) are the orbits with (cf. [3, Ch. II, Proposition 6.4], see also [2, Theorem]), whereas the classical ones are those with (see, e.g., [3, Ch. II, Proposition 6.3]).(v)In our discourse, the associated* ACP* may be* ill-posed*, i.e., the scalar type spectral operator need not generate a strongly continuous group of bounded linear operators (cf. [6]).

#### 2. Preliminaries

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

Henceforth, unless specified otherwise, is supposed to be a* scalar type spectral operator* in a complex Banach space with strongly -additive* spectral measure* (the* resolution of the identity*) assigning to each Borel set of the complex plane a projection operator on and having the operator’s* spectrum * as its* support* [7, 8].

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., [7, 8]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the* normal* ones [9].

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 [7, 8, 10, 11], which assigns to any Borel measurable function a scalar type spectral operator (see [7, 8]).

In particular,( is the set of* nonnegative integers*, , is the* identity operator* on ), andThe properties of the* spectral measure* and* operational calculus*, exhaustively delineated in [7, 8], underlie the entire subsequent discourse. Here, we underline a few facts of particular importance.

Due to its* strong countable additivity*, the spectral measure is* bounded* [8, 12], i.e., there is such an that, for any Borel set ,Observe that the notation is used here to designate the norm in the space of all bounded linear operators on . We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space .

For any and , the* total variation measure* of the complex-valued Borel measure is a* finite* positive Borel measure with(see, e.g., [13, 14]).

Also (Ibid.), for a Borel measurable function , , , and a Borel set ,In particular, for , andObserve that the constant in (15)–(17) is from (14).

Further, for a Borel measurable function , a Borel set , a sequence of pairwise disjoint Borel sets in , and , ,Indeed, since, for any Borel sets , [7, 8], for the total variation measure, Whence, due to the* nonnegativity* of (see, e.g., [15]),The following statement, allowing characterizing the domains of Borel measurable functions of a scalar type spectral operator in terms of positive Borel measures, is fundamental for our discourse.

Proposition 3 ([16, Proposition 3.1]). *Let be a scalar type spectral operator in a complex Banach space with spectral measure and be a Borel measurable function. Then iff *(i)*for each , ;*(ii)* , ,** where is the total variation measure of .*

*The succeeding key theorem provides a description of the weak solutions of (2) with a scalar type spectral operator in a complex Banach space.*

*Theorem 4 ([16, Theorem 4.2] with ). Let be a scalar type spectral operator in a complex Banach space . A vector function is a weak solution of (2) iff there is an such that the operator exponentials understood in the sense of the Borel operational calculus (see (13)).*

*Remark 5. *Theorem 4 generalizes [17, Theorem 3.1], its counterpart for a normal operator in a complex Hilbert space.

*We also need the following characterizations of a particular weak solution’s of (2) with a scalar type spectral operator in a complex Banach space being strongly differentiable on a subinterval of .*

*Proposition 6 ([1, Proposition 3.1] with ). Let and be a subinterval of . A weak solution of (2) is times strongly differentiable on iff in which case *

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

*3. General Weak Solution*

*Theorem 7 (general weak solution). Let be a scalar type spectral operator in a complex Banach space . A vector function is a weak solution of (1) iff there is an such thatthe operator exponentials understood in the sense of the Borel operational calculus (see (13)).*

*Proof. *As is noted in the Introduction, is a weak solution of (1)* iff*is a weak solution of (2) andis a weak solution of (8).

Applying Theorem 4, to and , we infer that this is equivalent to the fact

*Remarks 8. *(i)More generally, Theorem 4 and its proof can be easily modified to describe in the same manner all weak solution of (3) for an arbitrary interval of the real axis .(ii)Theorem 7 implies, in particular,(a)that the subspace of all possible initial values of the weak solutions of (1) is the largest permissible for the exponential form given by (25), which highlights the naturalness of the notion of weak solution;(b)that associated* ACP* (9), whenever solvable, is solvable* uniquely*.(iii)Observe that the initial-value subspace of (1), containing the dense in subspace , where which coincides with the class of* entire* vectors of of* exponential type* [18], is* dense* in as well.(iv)When a scalar type spectral operator in a complex Banach space generates a strongly continuous group of bounded linear operators, [6], and hence, Theorem 7 is consistent with the well-known description of the weak solutions for this setup (see (10)).(v)Clearly, the initial-value subspace of (1) is narrower than the initial-value subspace of (2) and the initial-value subspace of (8); in fact it is the intersection of the latter two.

*4. Differentiability of a Particular Weak Solution*

*Here, we characterize a particular weak solution’s of (1) with a scalar type spectral operator in a complex Banach space being strongly differentiable on a subinterval of *

*Proposition 9 (differentiability of a particular weak solution). Let and be a subinterval of . A weak solution of (1) is times strongly differentiable on iffin which case, *

*Proof. *The statement immediately follows from the prior theorem and Proposition 6 applied to for an arbitrary weak solution of (1).

*Remark 10. *Observe that, as well as for Proposition 6, for , the subinterval can degenerate into a singleton.

*Inductively, we immediately obtain the following analog of [1, Corollary 3.2].*

*Corollary 11 (infinite differentiability of a particular weak solution). Let be a scalar type spectral operator in a complex Banach space and be a subinterval of . A weak solution of (1) is strongly infinite differentiable on () iff, for each , in which case *

*5. Infinite Differentiability of Weak Solutions*

*In this section, we characterize the strong infinite differentiability on of all weak solutions of (1) with a scalar type spectral operator in a complex Banach space.*

*Theorem 12 (infinite differentiability of weak solutions). Let be a scalar type spectral operator in a complex Banach space with spectral measure . Every weak solution of (1) is strongly infinite differentiable on iff there exist and such that the set , where is bounded (see Figure 1).*