Let A be the infinitesimal generator  of a strongly continuous contraction semigroup in a  Hilbert space H.   We give an upper estimate for the best approximation of the operator A by bounded linear operators with a prescribed norm  in the space H on the class Q2={x∈D(A2):∥A2x∥≤1},  where D(A2) denotes the domain of A2.This work was supported by the Russian Foundation for Basic Research (project no. 15-01-02705), the Program for State Support of Leading Scientific Schools of the Russian Federation (project no.NSh-9356.2016.1), and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and the Ural Federal University)

Berdysheva, E. E.

Filatova, M. A.

Ural mathematical journal

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017ON THE BEST APPROXIMATION OF THE INFINITESIMALGENERATOR OF A CONTRACTION SEMIGROUPIN A HILBERT SPACE1Elena E. BerdyshevaDepartment of Mathematics, Justus Liebig University Giessen, Germany,elena.berdysheva@math.uni-giessen.deMaria A. FilatovaUral Federal University;Krasovskii Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences, Ekaterinburg, RussiaMA.Filatova@urfu.ruAbstract: Let A be the infinitesimal generator of a strongly continuous contraction semigroup in a Hilbertspace H. We give an upper estimate for the best approximation of the operator A by bounded linear operatorswith a prescribed norm in the space H on the class Q2 = {x ∈ D(A2) : ‖A2x‖ ≤ 1}, where D(A2) denotes thedomain of A2.Key words: Contraction semigroup, Infinitesimal generator, Stechkin’s problem.1. IntroductionLet H be a Hilbert space with the inner product (·, ·) and the norm ‖ · ‖, and let A be theinfinitesimal generator of a strongly continuous contraction semigroup in H. For the definitionand properties of the infinitesimal generator of a semigroup in a Banach space see, e.g., [6, §14.2].Note that a strongly continuous contraction semigroup is also called a contraction semigroup of theclass C0 ([8, 9]). For an operator F on the space H, D(F ) denotes the domain of F . We denoteby I the identity operator.In this paper, we study the so-called Stechkin’s problem of the best approximation of theoperator A by bounded linear operators with a prescribed norm on the class of elements x ∈ D(A2)such that ‖A2x‖ ≤ 1. We give an upper estimate for the best approximation of the operator A.The problem we consider is a special case of the general problem of the best approximation ofan unbounded operator by linear bounded ones on a certain class of elements in a Banach space.This problem first appeared in Stechkin’s work in 1965–1967 [11]. The problem was studied by anumber of authors (see surveys [1], [2], monograph [4], paper [3], and the bibliography therein).Stechkin formulated this problem in a general setting as follows. Let X, Y be two Banachspaces, let A be a linear operator (in general, unbounded) from X to Y , and let Q ⊆ D(A) be acertain class of elements from the domain D(A) of the operator A. We denote by B(N) the set of1This work was supported by the Russian Foundation for Basic Research (project no. 15-01-02705),the Program for State Support of Leading Scientific Schools of the Russian Federation (project no. NSh-9356.2016.1), and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August27, 2013, between the Ministry of Education and Science of the Russian Federation and the Ural FederalUniversity).Approximation of a generator of a contraction semigroup 41linear bounded operators from X to Y with the norm ‖T‖X→Y ≤ N . The best approximation ofthe operator A by linear bounded operators T ∈ B(N) on the class Q isEN (A;Q) = inf {U(A,T,Q) : T ∈ B(N)},whereU(A,T,Q) = sup {‖Ax − Tx‖Y : x ∈ Q}is the deviation of the operator T from the operator A on the class Q.One of the most important cases of the problem formulated above is when the class Q is definedin the following way. Let Z be a Banach space and B be a linear operator from X to Z such thatD(B) ⊆ D(A). The class Q is then defined as Q = {x ∈ X : ‖Bx‖Z ≤ 1}.Stechkin [11] suggested an estimate from below for the best approximation EN (A;Q) in termsof the modulus of continuity of the operator A on the class Q defined byΦ(δ) = sup {‖Ax‖Y : x ∈ Q, ‖x‖X ≤ δ}, δ > 0.Namely, Stechkin showed thatEN (A;Q) ≥ sup {Φ(δ) −Nδ : δ > 0}. (1.1)In particular, when B = An, the problem EN (Ak;Q) turned out to be closely connected to theexact constants in the Kolmogorov-type inequalities of the form‖Akx‖ ≤ C‖x‖n−kn ‖Anx‖ kn , x ∈ D(An), (1.2)with n, k ∈ N, 0 < k < n, and a certain constant C that depends on n and k.If A is the differentiation operator, inequalities (1.2) are inequalities between the norms ofthe derivatives of a function. Such inequalities have been studied by a large number of authors(see [1], [2], [4] and the bibliography therein). Here we only mention that Hardy, Littlewood andPólya [7, Chapter VII, §7.8] obtained the exact inequality‖f ′‖2 ≤ 2‖f‖‖f ′′‖ (1.3)in the space L2(0,∞) on the class of functions f ∈ L2(0,∞) such that f ′ is locally absolutelycontinuous on (0,∞), and f ′′ ∈ L2(0,∞).In 1971, Kato [9] proved the following result which can be considered as a generalization of (1.3).Let A be the infinitesimal generator of a strongly continuous contraction semigroup in a Hilbertspace H. Then‖Ax‖2 ≤ 2‖x‖‖A2x‖, x ∈ D(A2).In this paper, we study Stechkin’s problem of the best approximation of the infinitesimal gen-erator A of a strongly continuous contraction semigroup by bounded linear operators on the classQ2 = {x ∈ D(A2) : ‖A2x‖ ≤ 1} (1.4)in a Hilbert space. Namely, we estimateEN (A;Q2) = inf{U(T ) : T ∈ B(N)}, (1.5)whereU(T ) = U(A,T,Q2) = sup{‖Ax − Tx‖ : x ∈ Q2}. (1.6)42 E.E. Berdysheva and M.A. Filatova2. The main resultThe main result of the paper is the following statement.Theorem 1. The best approximation (1.5) of the infinitesimal generator A of a strongly con-tinuous contraction semigroup in a Hilbert space on the class Q2 defined in (1.4) satisfies theinequalityEN (A;Q2) ≤1N.It is known that the infinitesimal generator A of a strongly continuous contraction semigroupin a Banach space possesses the following properties:1) The domain D(A) of the operator A is dense (see, e.g., [6, Lemma 14.5, p. 411]).2) The resolvent set ρ(A) of the operator A contains the right half-plane {λ ∈ C| ℜλ > 0}.Moreover, ‖(A−λI)−1‖ ≤ (ℜλ)−1 for all λ ∈ C with ℜλ > 0 (e.g., [6, Theorem 14.7, p. 412]).Furthermore, if A is the infinitesimal generator of a strongly continuous contraction semigroupin a Hilbert space, we have additionally:3) The operator A is upper semibounded, with the upper bound 0, i.e.,ℜ(Ax, x) ≤ 0for x ∈ D(A) [6, Lemma 14.9, p. 416].The following lemma is not new. However, we will formulate and prove it for the sake ofcompleteness.Lemma 1. Let A be the infinitesimal generator of a strongly continuous contraction semigroupin a Hilbert space H and c > 0. Then the operatorBc = (cI +A)(cI −A)−1is densely defined and bounded (and thus can be extended to the whole space H by continuity).Moreover,‖Bc‖ ≤ 1.Remark. The operator Bc is the Cayley transform of the operator A in the terminology ofKato [9], see also [10, p. 545].P r o o f. Since c > 0, the operator (cI−A)−1 is defined everywhere on H and bounded. SinceA is the infinitesimal generator of a strongly continuous contraction semigroup, the operator −Ais m-accretive (see [10, Chapter IX, §1.4 as well as Problem 1.18, both p. 485]). Therefore, thedomain D(A) of the operator A is equal to the range R((cI − A)−1) of the operator (cI − A)−1which is dense in H (see [10, Chapter V, §3.10, p. 279]). Thus, Bc is densely defined.Now we estimate the norm of Bc. For x ∈ D(A) we have‖cx+Ax‖2 = c2‖x‖2 + ‖Ax‖2 + 2cℜ(Ax, x),Approximation of a generator of a contraction semigroup 43‖cx−Ax‖2 = c2‖x‖2 + ‖Ax‖2 − 2cℜ(Ax, x).It follows immediately that‖(cI +A)x‖ ≤ ‖(cI −A)x‖. (2.1)Now take y ∈ D((cI −A)−1). Applying (2.1) to x = (cI −A)−1y ∈ D(A), we obtain‖(cI +A)(cI −A)−1y‖ ≤ ‖y‖,and thus ‖Bc‖ ≤ 1. Now we are ready to prove Theorem 1.P r o o f. We will construct a concrete approximating operator T in problem (1.5) and estimateits norm and its deviation (1.6) from the operator A on the class Q2.Note that all the operators we consider commute on the set D(A2).The restriction of the operator A to the set D(A2) (which we will denote by the same symbol)can be represented asA =N2(BN − I)−12N(BN + I)A2.Put T : H → H,T =N2(BN − I).Then, for the restriction of the operator A− T to D(A2), we haveA− T = − 12N(BN + I)A2.We estimate the norm of the operator T as follows:‖T‖ = N2‖BN − I‖ ≤N2(‖BN‖+ ‖I‖) = N. (2.2)For the deviation U(T ) of the operator T from the operator A, we obtain thatU(T ) = supx∈Q2‖(A− T )x‖ ≤ supx∈Q212N‖BN + I‖ · ‖A2x‖ ≤1N. (2.3)It follows immediately from (2.2) and (2.3) thatEN (A;Q2) ≤ U(T ) ≤1N.3. Approximation of the differentiation operator in the space L2(0,∞)An important concrete case of problem (1.5) is the problem of the best approximation of thedifferentiation operator Df = f ′ by bounded linear operators in the Hilbert space L2(0,∞) ofreal-valued functions whose squares are integrable on (0,∞) on the class Q(2) defined as follows:44 E.E. Berdysheva and M.A. FilatovaQ(2) is the class of functions f ∈ L2(0,∞) such that f ′ is locally absolutely continuous on [0,∞),f ′′ ∈ L2(0,∞), and ‖f ′′‖ ≤ 1. Problem (1.5) takes in this case the formEN (D;Q(2)) = infT∈B(N)supf∈Q(2)‖f ′ − Tf‖. (3.1)It took about 20 years of research to solve the problem completely. Stechkin’s inequality (1.1) andinequality (1.3) of Hardy, Littlewood and Pólya provide the lower boundEN (D;Q(2)) ≥ 12N.One of the first upper bounds for (3.1)EN (D;Q(2)) ≤ 1√3Nwas obtained by using a concrete approximating operator by the first named author in 1996 [5].Problem (3.1) was fully solved only in 2014 by Arestov and the second named author [3] . Namely,they showed thatEN (D;Q(2)) =12N.In this section, we discuss what the statement of Theorem 1 means in the concrete case (3.1)of problem (1.5). The approximating operator T used in Theorem 1 isT =N2(BN − I) = NA(NI −A)−1. (3.2)Below we will describe this operator in the special case. We consider and calculate its norm ‖T‖and its deviation U(T ) from the operator A = D on the class Q(2).It is not difficult to see that the operator T in the concrete case can be represented as follows.Let W be the class of functions y ∈ L2(0,∞) such that y is locally absolutely continuous on [0,∞)and y′ ∈ L2(0,∞). For f ∈ L2(0,∞), we consider the differential equation−y′ +Ny = f, y ∈ W. (3.3)For each function f ∈ L2(0,∞), equation (3.3) has a unique solution which is a real-valued functionfrom L2(0,∞). The operator T is defined asTf = Ny′, (3.4)where y is the solution of the differential equation (3.3).Integrating by parts and taking into account that limt→∞y(t) = 0, we obtain (see [3] for details)that‖f‖2 =∞∫0(−y′(t) +Ny(t))2dt =∞∫0(y′(t))2dt+N2∞∫0(y(t))2dt+Ny2(0).It follows from (3.4) that ‖Tf‖2 = N2∞∫0(y′(t))2dt. Thus, we immediately obtain‖Tf‖2 ≤ N2‖f‖2, (3.5)Approximation of a generator of a contraction semigroup 45which gives the estimate ‖T‖ ≤ N . Now we show that indeed ‖T‖ = N . Consider the family offunctions yK = e−Kt, K > 0. Let fK be the corresponding right-hand side of equation (3.3). Takean arbitrary 0 < α < 1. We haveαN2‖fK‖2 − ‖TfK‖2 = αN2∞∫0(−y′K(t) +NyK(t))2dt−N2∞∫0(y′K(t))2dt=N22K(α(K +N)2 −K2).This expression is negative for all 0 < α <K2(N +K)2which yields ‖TfK‖2 > αN2‖fK‖2. Let-ting K go to infinity (with fixed N) we let α approach 1, and thus obtain ‖T‖ ≥ N . Conse-quently, ‖T‖ = N .Note that inequality (3.5) is a strict inequality if y 6= 0 and, consequently, f 6= 0. In otherwords, the norm of the operator T is not attained.It can be shown similarly that the norm of the operator V = − 12N(BN + I) is equal to 1/N .Since the domain D(D2) of the operator D2 is dense in L2(0,∞), it follows that the deviation ofthe operator T from the differentiation operator D on the class Q(2) is equal to 1/N .Thus, the approximating operator (3.2) gives the estimate EN (D;Q(2)) ≤ 1Nin the generalcase (1.5) as well as in the concrete case (3.1).REFERENCES1. Arestov V.V. Approximation of unbounded operators by bounded operators and related extremalproblems // Russian Math. Surveys, 1996. Vol. 51, no. 6. P. 1093–1126. DOI: 10.4213/rm10192. Arestov V.V., Gabushin V.N. Best approximation of unbounded operators by bounded operators //Russian Math. (Iz. VUZ), 1995. Vol. 39, no. 11. P. 38–63.3. Arestov V.V., Filatova M.A. Best approximation of the differentiation operator in the space L2 onthe semiaxis // J. Approx. Theory, 2014. Vol. 187. P. 65–81. DOI: 10.1016/j.jat.2014.08.0014. Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A. Inequalities for derivatives andtheir applications. Kiev: Naukova Dumka, 2003. [in Russian]5. Berdysheva E.E. On the best approximation of the differentiation operator in L2(0,∞) // East J.Approx. 1996. Vol. 2, no. 3. P. 281–287.6. Grubb G. Distributions and Operators. NY: Springer Science and Business Media, 2008. 464 p.DOI: 10.1007/978-0-387-84895-27. Hardy G.H., Littlewood J.E., Pólya G. Inequalities. Cambridge University Press, 1934. 314 p.8. Hille E., Phillips R.S. Functional Analysis and Semigroups. American Mathematical Society, 1957.808 p.9. Kato T. On an inequality of Hardy, Littlewood and Pólya // Adv. Math., 1971. Vol. 7. P. 217–218.DOI: 10.1016/S0001-8708(71)80002-610. Kato T. Perturbation Theory for Linear Operators. Verlag–Berlin–Heidelberg: Springer, 1995. 623 p.DOI: 10.1007/978-3-642-66282-911. Stechkin S.B. Best approximation of linear operators // Math. Notes. 1967. Vol. 1. P. 91–99.DOI: 10.1007/BF01268056

On the Best Approximation of the Infinitesimal Generator of a Contraction Semigroup in A Hilbert Space

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On the Best Approximation of the Infinitesimal Generator of a Contraction Semigroup in A Hilbert Space

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