SOME APPLICATIONS OF FEJER'S THEOREM TO OPERATOR ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 8, August 1997, Pages 2353–2362 S 0002-9939(97)03837-9

SOME APPLICATIONS OF FEJER’S THEOREM TO OPERATOR COSINE FUNCTIONS IN BANACH SPACES ˘ IOANA CIORANESCU AND CARLOS LIZAMA (Communicated by Palle E. T. Jorgensen)

Abstract. We characterize spectral properties of operator cosine functions in Banach spaces in terms of the C´ esaro summability of two series associated to the resolvent of the corresponding infinitesimal generator.

1. Introduction Given a strongly continuous cosine family of linear and bounded operators (C(t))t∈R acting on a complex Banach space X with generator A, the following spectral inclusion p cosh t σ(A) ⊂ σ(C(t)), t ∈ R, was obtained by B. Nagy; he also proved that the reverse inclusion may fail quite drastically (see [9]). In particular −N02 ⊂ ρ(A) is implied by 1 ∈ ρ(C(2π)) but not conversely. In the case of a Hilbert space, I. Cioranescu and C. Lizama [1] proved the following result: (∗)

1 ∈ ρ(C(2π)) if and only if − N02 ⊂ ρ(A)

and Sup||kR(−k 2 ; A)|| < ∞; k∈Z

however, the problem of the validity of (∗) in general Banach spaces was left open. Results of this type for C0 - semigroups, as well as applications, were obtained by different authors (see [8], [6] and the references therein). Recently, G. Greiner and M. Schwarz [4] (see also G. Greiner [3] ) have proved a spectral mapping theorem in the Banach space setting involving the C´esaro summability for C0 - semigroups P of the series k R(ik; A). Their approach is based on the following vector valued version of the classical result due to Fejer [5]. Theorem 1.1. Let X be a Banach space, f : [0, 2π] → X a continuous function R 2π −iks 1 and zk := 2π e f (s)ds its k-th Fourier coefficient; then the Fourier series of 0 Received by the editors December 11, 1995 and, in revised form, February 22, 1996. 1991 Mathematics Subject Classification. Primary 47D09; Secondary 34G10. This research was done while the first author was visiting the Department of Mathematics, University of Santiago de Chile, supported by CONICYT. The second author was partially supported by FONDECYT grant 1930066 and DICYT (USACH). c

1997 American Mathematical Society

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˘ IOANA CIORANESCU AND CARLOS LIZAMA

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f is C´esaro summable to f (t) in every point t ∈ (0, 2π). Moreover, one has C1 −

X

N −1 n f (0) + f (2π) 1 X X . zk = N →∞ N 2 n=0

zk := lim

k∈Z

k=−n

” The aim of this work is to characterize “1 in terms of P∈ ρ(C(2π)) P the property 2 2 2 the C´esaro summability of the series k R(−k ; A) and k AR (−k ; A). We also show, in section 3, that our main theorem reduces to (∗) in the Hilbert space case. We remark that for operator cosine functions the behaviour of the single spectral value 1 of C(2π) fails to characterize σ(C(t)) since the usual rescaling procedure, from the theory of C0 -semigroups (see [4], [8]), doesn’t work. Finally, in the last section, applications are made to prove the existence of solutions of some second order differential equations in Banach spaces with Dirichlet or Cauchy conditions.

2. Main result The main result in this section is the following: Theorem 2.1. Let (C(t)) be an operator cosine function on the Banach space X, A its generator and (S(t)) the associated sine function; then the following assertions are equivalent: a) b)

1 ∈ ρ (C(2π)) . −N02 ⊂ ρ(A) and the sequences P −1 Pn 2 RN = N1 N n=0 k=−n R(−k ; A) SN =

and

N −1 n 1 X X AR2 (−k 2 ; A) N n=0 k=−n

c)

are bounded in L(X). −N02 ⊂ ρ(A) and the limits Rx = lim RN x N →∞

and

Sx = lim SN x N →∞

exist for every x ∈ X. Proof. a) ⇒ b) In [9] the following formula was proved: Z t 2 (a − A) sinh a(t − s)C(s)xds (1) 0 = a[(cosh at)x − C(t)x], x ∈ X, a ∈ C, t ∈ R. Integrating by parts and then taking the derivative with respect to t we obtain the following identity: Z t (2) cosh a(t − s)S(s)xds = (cosh at)x − C(t)x (a2 − A) 0

for every x ∈ X and a ∈ C. We now take t = 2π and a2 = λ to obtain Z 2π √ √ (3) cosh λ(s − 2π)S(s)xds = (cosh 2π λ)x − C(2π)x. (λ − A) 0

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SOME APPLICATIONS OF FEJER’S THEOREM

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√ Assuming that cosh 2π λ ∈ ρ (C(2π)), we have Z 2π  √ √ R(λ; A) = (4) cosh λ(s − 2π)S(s)ds [cosh 2π λ − C(2π)]−1 . 0

For the strongly continuous operator-valued function K on [0, 2π], we denote Z 2π  Z 2π K(s)ds the linear and continuous operator defined by K(s)ds x := by 0 Z 2π 0 K(s)xds, x ∈ X. 0

From (4) one easily obtains d R(λ; A) dλ Z 2π √ √ 1 = √ (s − 2π) sinh λ(s − 2π)S(s)[cosh 2π λ − C(2π)]−1 ds 2 λ 0 √ Z 2π √ √ π sinh 2π λ √ cosh λ(s − 2π)S(s) [cosh 2π λ − C(2π)]−2 ds. − λ 0

−R2 (λ; A) =

(5)

If 1 ∈ ρ(C(2π)), then by the spectral mapping theorem for operator cosine functions [9] we have −N02 ⊂ ρ(A). Then for λ = −k 2 , k ∈ Z, (4) and (5) respectively yield Z 2π  R(−k 2 ; A) = (6) cos ksS(s)ds [I − C(2π)]−1 0

and −R2 (−k 2 ; A) =

1 2k

Z

2π 0

 (s − 2π) sin ksS(s)ds [I − C(2π)]−1 .

It follows that (on D(A) and therefore on all X) we have Z 2π  1 2 2 − AR (−k ; A) = (s − 2π) sin ksAS(s)ds [I − C(2π)]−1 2k 0 Z 2π  1 0 = (s − 2π) sin ksC (s)ds [I − C(2π)]−1 2k 0 Z 2π  Z 2π 1 =− k(s − 2π) cos ksC(s)ds + sin ksC(s)ds [I − C(2π)]−1 2k 0 0 Z 2π  Z 2π 1 sin ks 0 S (s)ds [I − C(2π)]−1 =− (s − 2π) cos ksC(s)ds + 2 0 k 0 Z 2π  Z 2π 1 =− (s − 2π) cos ksC(s)ds − cos ksS(s)ds [I − C(2π)]−1 . 2 0 0 So that we finally obtain Z 2π  1 2 2 (7) AR (−k ; A) = cos ks[(s − 2π)C(s) − S(s)]ds [I − C(2π)]−1 . 2 0

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˘ IOANA CIORANESCU AND CARLOS LIZAMA

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By (6) we have RN =

N −1 n N −1 n Z 2π 1 X X 1 X X R(−k 2 ; A) = cos ksS(s)[I − C(2π)]−1 ds N n=0 N n=0 0 k=−n

k=−n

=

so that

Z

(8)

RN =

0

2π

1 N

N −1 X

n Z X

n=0 k=−n

2π

0

e−iks S(s)[I − C(2π)]−1 ds

σN (s)S(s)[I − C(2π)]−1 ds

PN −1 Pn −iks where σN (s) = N1 n=0 = k=−n e Once again by (7) we obtain

1 1−cos N s N 1−cos s

≥0

and

R 2π 0

σN (s)ds = 2π.

N −1 n 1 X X AR2 (−k 2 ; A) N n=0 k=−n Z 2π (s − 2π)C(s) − S(s) [I − C(2π)]−1 ds. σN (s) = 2 0

SN = (9)

Now (8) and (9) yield kRN k ≤ 2πM1 k[I − C(2π)]−1 k and kSN k ≤ 2πM2 k[I − C(2π)]−1 k with M1 = Sup{kS(s)k; 0 ≤ s ≤ 2π} and

 M2 = Sup

 ||(s − 2π)C(s) − S(s)|| ; 0 ≤ s ≤ 2π . 2

b) ⇒ c) Let −N02 ⊂ ρ(A); then by the spectral mapping theorem for the residual spectrum for operator cosine functions (see [9]) it follows that 1 ∈ / σr (C(2π)); this implies that (I − C(2π))X is a dense subset of X. For y ∈ X, let x = (I − C(2π))y; then by the same arguments that led to (8) and (9) we have RN x =

N −1 n 1 X X R(−k 2 ; A)(I − C(2π))y N n=0 k=−n

(10)

=

N −1 n Z 2π 1 X X e−iks S(s)yds N n=0 0 k=−n

and SN x =

N −1 n 1 X X AR2 (−k 2 ; A)(I − C(2π))y N n=0 k=−n

(11)

N −1 n Z 2π 1 X X (s − 2π)C(s) − S(s) yds. = e−iks N n=0 2 0 k=−n

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SOME APPLICATIONS OF FEJER’S THEOREM

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By Fejer’s theorem, it follows that the limits Rx = limN →∞ RN x and Sx = limN →∞ SN x exist for x in a dense subset of X. Since the sequences (RN ) and (SN ) are bounded in L(X), the limits exist for every x ∈ X. c) ⇒ a) Using again Fejer’s theorem and letting N → ∞, then from (10) and (11) we obtain R(I − C(2π)) = πS(2π) and 2S(I − C(2π)) = −2π 2 I − πS(2π). Now, it follows that (R + 2S)(I − C(2π)) = −2π 2 I. Hence − 2π1 2 (R + 2S) is the inverse of I − C(2π). (We notice that R + 2S commutes with I − C(2π).) In [2] the following class of operators was considered in connection with second order differential equations of elliptic type Definition 2.2. A linear operator A defined in a Banach space X is called apositive (a > 0) if π2 k2 −π 2 N02 2 ⊂ ρ(A) and Sup ||k R(− ; A)|| < ∞. k∈Z a2 a2 Proposition 2.3. If A is π-positive then the sequences RN =

N −1 n 1 X X R(−k 2 ; A) N n=0

and

k=−n

SN =

N −1 n 1 X X AR2 (−k 2 ; A) N n=0 k=−n

are bounded in L(X). Proof. We observe that by hypothesis there is a constant M > 0 such that M , k ∈ Z, k 6= 0, k2 PN −1 Pn = N1 n=0 n=−n R(−k 2 ; A) follows. On the

kR(−k 2 ; A)k ≤ from which the boundedness of RN other hand, we may write SN =

N −1 n N −1 n 1 X X 1 X X AR2 (−k 2 ; A) = R(−k 2 ; A)AR(−k 2 ; A) N n=0 N n=0 k=−n

k=−n

where kAR(−k 2 ; a)k = k − I + k 2 R(−k 2 ; A)k ≤ 1 + M . So we also obtain the boundedness of SN . 3. The Hilbert space case We shall show that in the case of a Hilbert space we can derive from Theorem 2.1 the spectral mapping theorem from [1] (see also [7]) in which the summability conditions stated in (c) can be replaced by some boundedness conditions on the resolvents. For this purpose we shall need the following lemma which can be proved as for scalar-valued functions. Lemma 3.1. Let H be a Hilbert space. Assume f : [0, 2π] → H is continuous R 2π 1 f (s)e−iks ds are (square integrable). Then the Fourier coefficients zk := 2π 0 square summable; more precisely, Z 2π ∞ X 1 2 ||zk || = ||f (s)||2 ds. 2π 0 k=−∞

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˘ IOANA CIORANESCU AND CARLOS LIZAMA

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Pn Conversely, if {zk } is a sequence in H and fn := k=−n zk eiks , then Z 2π n X 1 2 ||zk || = ||fn (s)||2 ds. 2π 0 k=−n

Theorem 3.2. Let (C(t)) be an operator cosine function on the Hilbert space H, A its generator and (S(t)) the associated sine function. Then the following assertions are equivalent: (i) 1 ∈ ρ (C(2π)) . (ii) −N02 ⊂ ρ(A) and Supk∈Z kkR(−k 2 ; A)k < ∞. Proof. (i)⇒(ii) It is an easy consequence of formula (1). √ Then replacing in (ii)⇒(i) Suppose λ ∈ C is √ such that cos 2π λ ∈ ρ (C(2π)). √ formula (2) t = 2π, a = τk := λ + ik and T := [cosh 2π λ − C(2π)]−1 we have Z 2π  cosh τk (s − 2π)S(s)ds T x R(τk2 ; A)x = 0

=

1 2 Z

Z

2π

0 2π

e

= 0

eiks e

iks

√ λ(s−2π)

S(s)T xds +

Z F (s)xds +

2π

0

1 2

Z

2π

0

e−iks e−

√ λ(s−2π)

S(s)T xds

e−iks G(s)xds

√

√

where F (s) = 12 e λ(s−2π) S(s)T, G(s) = 12 e− λ(s−2π) S(s)T and x is fixed in H . Then by Lemma 3.1, the sequence {uk } := R(τk2 ; A)x k∈Z is square summable. We also have from (1) with t = 2π, a = τk and T as above that Z 2π 2 sinh τk (2π − s)C(s)T xds τk R(τk ; A)x = 0

1 = 2 Z =

Z 0 2π

0 √

2π

eiks e

√ λ(2π−s)

eiks H(s)xds −

C(s)T xds −

Z 0

2π

1 2

Z

2π

0

e−iks e−

√ λ(2π−s)

C(s)T xds

e−iks L(s)xds √

where H(s) = 12 e λ(2π−s) C(s)T, L(s) = 12 e− λ(2π−s) C(s)T . It follows that the sequence√{vk } := {τk R(τk2 ; A)x}k is square summable. Let zk := kR(τk2 ; A)x = i( λuk − vk ); then it is clear that {zk } is also square summable. From the resolvent equation we have √ R(−k 2 ; A) = R(τk2 ; A)[I + (λ + 2ik λ)R(−k 2 ; A)] (12) √ 2 ; A)x = [I + (λ + 2ik λ)R(−k 2 ; A)]uk and kR(−k 2 ; A)x = so that R(−k √ 2 [I + (λ + 2ik λ)R(−k ; A)]zk √ Since by hypothesis Supk∈Z k(λ + 2ik λ)R(−k 2 ; A)k < ∞ , it follows that the sequences {R(−k 2 ; A)x} and {kR(−k 2 ; A)x} are √ square summable. Finally we get that the sequence {wk }k where wk = (λ + 2ik λ)R(−k 2 ; A)x is square summable. We shall prove that the conditions of the Theorem 2.1 (b) are satisfied.

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SOME APPLICATIONS OF FEJER’S THEOREM

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Using (12) we have N −1 n N −1 n N −1 n 1 X X 1 X X 1 X X R(−k 2 ; A)x = R(τk2 ; A)x + R(τk2 ; A)wk . N n=0 N n=0 N n=0 k=−n

k=−n

k=−n

We estimate the norm of the first term in the above sum: N −1 n N −1 n Z 2π 1 X X 1 X X 2 k R(τk ; A)xk = k [eiks F (s)x + e−iks G(s)x]dsk N n=0 N n=0 0 k=−n k=−n Z 2π σN (s)(F (s)x + G(s)x)ds|| = || 0 2π

Z ≤

0

σN (s)||F (s)x + G(s)x||ds Z

≤ Sups∈[0,

2π] kF (s)x + G(s)xk.

2π 0

σN (s)ds < ∞.

For the second term we have

n Z

n

X

X 2π



2 iks −iks R(τk ; A)wk = (e F (s)wk + e G(s)wk )ds



0 k=−n

k=−n

Z ≤

2π 0

Z + 0

2

1/2

kF (s)k ds

2π

kG(s)k2 ds

Z

2π

0

k

n X

!1/2 e

iks

2

wk k ds

k=−n

1/2 Z

2π 0

k

n X

!1/2 e−iks wk k2 ds

.

k=−n

Using Lemma 3.1 we obtain that the second term is also bounded. From (12) we also have that √ AR2 (−k 2 ; A) = AR2 (τk2 ; A)[I + (λ + 2ik λ)R(−k 2 ; A)]2 √ = [−R(τk2 ; A) + τk2 R2 (τk2 ; A)][I + (λ + 2ik λ)R(−k 2 ; A)]2 √ √ = −R(τk2 ; A) − R(τk2 ; A)[2(λ + ik λ)R(−k 2 ; A) + (λ + 2ik λ)2 R2 (−k 2 ; A)] √ + τk R(τk2 ; A){τk R(τk2 ; A)[I + (λ + 2ik λ)R(−k 2 ; A)]2 }. Denote

√ √ xk = 2(λ + ik λ)R(−k 2 ; A)x + [(λ + 2ik λ)R(−k 2 ; A)]2 x, √ yk = τk R(τk 2 ; A)[I + (λ + 2ik λ)R(−k 2 ; A)]2 x.

Then {xk } and {yk } are square summable.

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˘ IOANA CIORANESCU AND CARLOS LIZAMA

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We have AR2 (−k 2 ; A)x = −R(τk 2 ; A)x − R(τk 2 ; A)xk + τk R(τk 2 ; A)yk so that N −1 n N −1 n N −1 n 1 XX 1 XX 1 X X AR2 (−k 2 ; A)x = − R(τk 2 ; A)x − R(τk 2 ; A)xk N n=0 N n=0 −n N 0 −n k=−n

+

N −1 n 1 XX τk R(τk 2 , A)yk . N 0 −n

The first sum was estimated above; the second one can be exactly worked as was done previously and for the third one we have practically the same, since n n Z 2π X X τk R(τk2 ; A)yk k = k (eiks H(s) + e−iks L(s))yk dsk k k=−n

k=−n

Z ≤

2π

0

Z

||H(s)||2 ds

2π

+ 0

2

1/2

0

Z

2π

0

!1/2 eiks yk k2 ds

k=−n

1/2 Z

||L(s)|| ds

k

n X

2π 0

k

!1/2

n X

e

−iks

2

yk k ds

< ∞.

k=−n

4. Applications In this section, we first consider the following Dirichlet problem  00 0 < t < π,  u (t) = Au(t), (4.1) u(0) = x,  u(π) = y, where x, y ∈ D(A), A is a linear densely defined closed operator on a Banach space X and u : (0, π) → D(A) is a twice continuously differentiable function. Theorem 4.1. Let A be the generator of an operator cosine function (C(t)) on the Banach space X, and let (S(t)) be the associated sine function. Suppose that −N02 ⊂ ρ(A) and that both limits Rx = lim RN x N →∞

and

Sx = lim SN x N →∞

exist for every x ∈ X. Then, there is a unique solution of the Dirichlet Problem (4.1). Moreover, there is a constant c such that (13)

Sup{||u(s)|| : 0 ≤ s ≤ π} ≤ c(ku(0)k + ku(π)k).

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Proof. From Theorem 2.1 it follows, under our hypothesis, that the operator (I − C(2π)) is invertible. Next, from the identity (see [10] Prop. 2.2) C(t + s) − C(t − s) = 2AS(t)S(s) taking t = s = π we obtain that S(π) is an invertible operator. Define u(t) = S(t)S(π)

−1

−1

y + S(π − t)S(π)

x.

Then u(t) is the (unique) solution of (4.1) and the result follows. Corollary 4.2. If A is π-positive and generates an operator cosine function, then the Dirichlet Problem (4.1) has a unique solution satisfying (13). Finally, we consider the second order abstract Cauchy problem  00 t ∈ R,  u (t) = Au(t) + f (t), (4.2) u(0) = x,  0 u (0) = y. We remark that if f ∈ C 1 (R, X) and (x, y) ∈ D(A) × E , where E = {x ∈ X/t → C(t)x is once continuously differentiable }, then the classical solutions of (4.2) are given by the formula Z t (14) S(t − s)f (s)ds, t ∈ R. u(t) = C(t)x + S(t)y + 0

L2loc (R, X),

If (x, y) ∈ X × X and f ∈ we call (14) a mild solution of (4.2). It is clear that this mild solution is of class C 1 if and only if (x, y) ∈ E × X. The following result was proved in [1]. Theorem 4.3. Let A be the generator of a strongly continuous cosine function (C(t)) in the Banach space X; then 1 ∈ ρ(C(2π)) if and only if for any 2π-periodic function f ∈ L2loc (R, X), the equation (4.2) has a unique 2π-periodic mild solution of class C 1 . Combining Theorems 2.1 and 4.3 we obtain the following Corollary 4.4. Let X be a Banach space and A the generator of a strongly continuous cosine function (C(t)) in X; the following assertions are equivalent: i) The equation (4.2) has a unique 2π-periodic mild solution of class C 1 for any 2π - periodic function f ∈ L2loc (R, X). ii) −N02 ⊂ ρ(A) and both limits Rx = lim RN x N →∞

and

Sx = lim SN x N →∞

exist for every x ∈ X. References 1. I. Cioranescu, C. Lizama, Spectral properties of cosine operator functions, Aequationes Math. 36 (1988) 80-98. MR 89i:47071 2. V.I. Gorbachuk, A.V. Knyazyuk, Boundary values of solutions of operator-differential equations, Russian Math. Surveys 44:3 (1989) 67-111. MR 91c:47080 3. G. Greiner, A short proof of Gearhart’s theorem, Semesterbericht Funktionalanalysis Tubingen SS 89 16 (1989) 89-92. 4. G. Greiner, M. Schwarz, Weak spectral mapping theorems for functional differential equations, J.Diff. Eq. 94 (1991) 205-216. MR 92k:47079 5. Y. Katznelson, Harmonic Analysis, New York-London-Sydney-Toronto, 1968. MR 40:1734

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˘ IOANA CIORANESCU AND CARLOS LIZAMA

6. Y. Latushkin, S. Montgomery-Smith, Lyapunov theorems for Banach spaces, Bull. Amer. Math. Soc. (New series) 31 (1994) 44-49. MR 94j:47062 7. C. Lizama, On the spectrum of cosine operator functions, Int. Eq. Operator Theory 12 (1989) 713-724. MR 90h:47074 8. R.Nagel, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, Vol. 1184, Springer-Verlag, Berlin/New York, 1980. MR 88i:47022 9. B. Nagy, On cosine operator functions in Banach spaces, Acta Sci. Math. (Szeged) 36 (1974) 281-290. MR 51:11191 10. C.C. Travis, G.F. Webb, Cosine families and abstract non-linear second order differential equations, Acta Math. Acad. Sci. Hung. 32 (1978) 75-96. MR 58:17404 Department of Mathematics, University of Puerto Rico, Box 23355 Rio Piedras, Puerto Rico 00931 ´ tica y C. C., Universidad de Santiago de Chile, Casilla Departamento de Matema 307-Correo 2, Santiago, Chile E-mail address: [email protected]

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