A factorization of a quadratic pencils of accretive operators and applications
aa r X i v : . [ m a t h . F A ] A ug A FACTORIZATION OF A QUADRATIC PENCILS OF ACCRETIVEOPERATORS AND APPLICATIONS
FAIROUZ BOUCHELAGHEM , MOHAMMED BENHARRAT ∗ Abstract.
A canonical factorization is given for a quadratic pencil of accretive oper-ators in a Hilbert space. Also, we establish some relationships between an m-accretiveoperator and its Moore-Penorse inverse. As an application, we study a result of exis-tence, uniqueness, and maximal regularity of the strict solution for complete abstractsecond order differential equation in the non-homogeneous case. The paper is concludedwith some questions left open from the preceding discussions. introduction Many problems in mathematical physics and mechanics can be described by the follow-ing second order linear differential equation u ′′ ( t ) − Bu ′ ( t ) − Cu ( t ) = 0 , (1.1)where u ( t ) is a vector-valued function in an appropriate (finite or infinite dimensional)Hilbert space H , B and C are linear (bounded or unbounded) operators on H . Proper-ties of the differential equation (1.1) are closely connected with spectral properties of aquadratic pencil Q ( λ ) = λ I − λB − C, ( λ ∈ C ); (1.2)which is obtained by substituting exponential functions u ( t ) = exp( λt ) x , x ∈ H into(1.1). In many applications B and C are self-adjoint positive definite operators. Animportant and subtle problem in the theory of such operator pencils is to factoring themand studying the spectral properties of the factors. Krein and Langer [13] proved that aself-adjoint polynomial of the form (1.2) can always be written as a product of two linearfactors as follows λ I − λB − C = ( λI − Z )( λI − Z ) , (1.3)with Z and Z are a roots of the quadratic operator equation Q ( Z ) = Z I − BZ − C = 0 . (1.4)Of particular interest is the separation of spectral values of Q between the spectra ofthe roots. Such separation may be complicated, even in the case of eigenvalues, see [23]and references therein. The factorization theorems have been studied extensively also forthe self-adjoint quadratic operator pencils under the extra condition of strong and weak Date : 29/07/2020. ∗ Corresponding authorThis work was supported by the Laboratory of Fundamental and Applicable Mathematics of Oran (LM-FAO) and the Algerian research project: PRFU, no. C00L03ES310120180002.2010
Mathematics Subject Classification.
Primary 47A10; 47A56.
Key words and phrases.
Quadratic operator pencil, Spectral theory, Accretive operators, Semigroupof contraction. damping. For the exhaustive survey on these topics, please see the two seminal books [17]and [18] and the references therein.But some models of continuous mechanics are reduced to differential equation (1.1)with sectorial operators, see [1, 3, 8, 12] and references therein. In this cases methods,developed for self-adjoint operators, cannot be applied.In this paper we are going to study a class of non-self-adjoint quadratic operator pencilsthe coefficients of which are unbounded accretive operators. Our aim is to investigate acanonical factorization like (1.3) for of such pencils based on the perturbation theory ofaccretive operators. We also obtain a criterion in order that the linear factors, into whichthe pencil splits, generates a strongly continuous semi-group of contraction operators. Weapply this result to establish a theorem of existence, uniqueness, and maximal regularityof the strict solution of an abstract second order evolutionary equations generated by suchpencils in the non-homogeneous case. The paper is then concluded with a summary ofthe problems left open from the preceding discussions.2.
Accretive operators framework
In this section we introduce the notation and the operator theoretic framework usedin the rest of our work. Throughout this paper H is a complex Hilbert space with innerproduct < · , · > and norm k · k . Let B ( H ) denote the Banach space of all boundedlinear operators on H . Given a linear operator T on H we denote by D ( T ), N ( T ), and R ( T ) the domain, the null space and the range of T , respectively. For a closable denselydefined linear operator T in some Hilbert space H we denote by ρ ( T ) the resolvent set,by σ ( T ) = C \ ρ ( T ) the spectrum, and by σ p ( T ) the point spectrum of T . For λ ∈ ρ ( T ) , the inverse ( λI − T ) − is, by the closed graph theorem, a bounded operator on H and willbe called the resolvent of T at the point λ .Recall that a linear operator T with domain D ( T ) in a complex Hilbert space H is saidto be accretive if Re < T x, x > ≥ x ∈ D ( T )or, equivalently if k ( λ + T ) x k ≥ λ k x k for all x ∈ D ( T ) and λ > . An accretive operator T is called maximal accretive , or m -accretive for short, if one of thefollowing equivalent conditions is satisfied:(1) T has no proper accretive extensions in H ;(2) T is densely defined and R ( λ + T ) = H for some (and hence for every) λ > T is densely defined and closed, and T ∗ is accretive;(4) − T generates contractive one-parameter semigroup T ( t ) = exp( − tT ), t ≥ λ + T ) − ∈ B ( H ) and (cid:13)(cid:13) ( λ + T ) − (cid:13)(cid:13) ≤ λ for λ > . In particular, a bounded accretive operator is m-accretive. ˆOta showed in [22, Theorem2.1] that, if T is closed and an accretive such that there is a positive integer n with D ( T n )is dense in H and R ( T n ) ⊂ D ( T ), then T is bounded . In particular, for a closed and FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 3 accretive operator T , if R ( T ) is contained in D ( T ), or in D ( T ∗ ), then T is automaticallybounded, see also [22, Theorem 3.3]. Also, if T is maximal accretive, then N ( T ) = N ( T ∗ ) and N ( T ) ⊆ D ( T ) ∩ D ( T ∗ ) . (2.1)The numerical range is very useful set by what we can we define the accretive operators.For a linear operator T : D ( T ) → H it is defined by W ( T ) := { < T x, x > : x ∈ D ( T ) , with k x k = 1 } , (2.2)It is well-known that W ( T ) is a convex set of the complex plane (the Toeplitz-Hausdorfftheorem), and in general is neither open nor closed, even for a closed operator T . Clearly,an operator T is accretive when W ( T ) is contained in the closed right half-plane W ( T ) ⊂ C + := { z ∈ C : Re(z) ≥ } . Further, if T is m-accretive operator then W ( T ) has the so-called spectral inclusion prop-erty σ ( T ) ⊂ W ( T ) . (2.3)A linear operator T in a Hilbert space H is called sectorial with vertex z = 0 andsemi-angle ω ∈ [0 , π/ ω -accretive for short, if its numerical range is contained in aclosed sector with semi-angle ω , W ( T ) ⊂ S ( ω ) := { z ∈ C : | arg z | ≤ ω } (2.4)or, equivalently, | Im < T x, x > | ≤ tan ω Re < T x, x > for all x ∈ D ( T ) . An ω -accretive operator T is called m- ω -accretive, if it is m -accretive. We have T ism- ω -accretive if and only if the operators e ± iθ T is m-accretive for θ = π − ω , 0 < ω ≤ π/ ω -accretive operator T contains the set C \ S ( ω ) and k ( T − λI ) − k ≤ λ, S ( ω )) , λ ∈ C \ S ( ω ) . In particular, m- π/ C -semigroup T ( t ) = exp( − tT ), t ≥
0, has contractive andholomorphic continuation into the sector S ( π/ ω ) if and only if the generator T is m- ω -accretive, see [14, Theorem V-3.35].Recently, the authors of [2] obtained a precise localization of the numerical range of one-parameter semigroup T ( t ) = exp( − tT ), t ≥
0, generated by an m- ω -accretive operator, ω ∈ [0 , π/ W (exp( − tT )) ⊆ Ω( ω ) = { z ∈ C : (cid:12)(cid:12) Im √ z (cid:12)(cid:12) ≤
12 (1 − | z | ) tan( ω ) } , t ≥ , (2.5)with limiting cases: Ω(0) = [0 ,
1] and Ω( π/
2) = D . In particular, the family exp( − tT ) t ≥ is a quasi-sectorial contractions semigroup in the terminology of [2].We mention that if T is m -accretive, then for each α ∈ (0 ,
1) the fractional powers T α ,0 < α <
1, are defined by the following Balakrishnan formula, see [4], T α x = sin( πα ) π Z ∞ λ α − T ( λ + T ) − xdt, F. BOUCHELAGHEM, M. BENHARRAT for all x ∈ D ( T ). The operators T α are m-( απ ) / α ∈ (0 , / D ( T α ) = D ( T ∗ α ). It was proved in [15, Theorem 5.1] that, if T is m -accretive, then D ( T / ) ∩ D ( T ∗ / ) is a core of both T / and T ∗ / and the real part ReT / := ( T / + T ∗ / ) / D ( T / ) ∩ D ( T ∗ / ) is a selfadjoint operator. Further, by [15, Corol-lary 2], D ( T ) = D ( T ∗ ) = ⇒ D ( T / ) = D ( T ∗ / ) = D ( T / R ) = D [ φ ] , (2.6)where φ is the closed form associated with the sectorial operator T via the first representa-tion theorem [14, Sect. VI.2.1] and T R is the non-negative selfadjoint operator associatedwith the real part of φ given by Re φ := ( φ + φ ∗ ) / Accretive operator and the Moore-Penrose inverse
Next, in order to give some new results about accretive operator by using the Moore-Penrose inverse, let recall the definition of this generalized inverse for a closed denselydefined operator.
Definition 3.1. [5] Let T be a closed densely defined on H . Then there exists a uniqueclosed densely defined operator T † , with domain D ( T † ) = R ( T ) ⊕ R ( T ) ⊥ such that T T † T = T on D ( T ) , T † T T † = T † on D ( T † ) ,T T † = P R ( T ) on D ( T † ) , T † T = P N ( T ) ⊥ on D ( T ) , with P M denotes the orthogonal projection onto a closed subspace M .This unique operator T † is called the Moore-Penrose inverse of T . (or the MaximalTseng generalized Inverse in the terminology of [5]). Clearly,(1) N ( T † ) = R ( T ) ⊥ ,(2) R ( T † ) = N ( T ) ⊥ ∩ D ( T ).As a consequence of the closed graph theorem T † is bounded if and only if R ( T ) is closedin H , see [5].Now, if we assume that T is an m-accretive operator, by (2.1), N ( T ) = N ( T ∗ ) andthus R ( T ) = R ( T ∗ ) and H = R ( T ) ⊕ N ( T ). Consequently, the operator T is written ina matrix form with respect to mutually orthogonal subspaces decomposition as follows T = (cid:20) T
00 0 (cid:21) : (cid:20) R ( T ) N ( T ) (cid:21) −→ (cid:20) R ( T ) N ( T ) (cid:21) ;with T is an operator on R ( T ) ∩ D ( T ) is injective with dense range in R ( T ). Also, itsMoore-Penrose inverse is given by T † = (cid:20) T −
00 0 (cid:21) : (cid:20) R ( T ) N ( T ) (cid:21) −→ (cid:20) R ( T ) N ( T ) (cid:21) , with T − from R ( T ) to R ( T ) ∩ D ( T ) is closed operator densely defined on R ( T ) and N ( T † ) = N ( T ) = N ( T ∗ ). Further, R ( T ) is closed if and only if T − is bounded from R ( T ) to R ( T ) ∩ D ( T ). Proposition 3.2. If T is m-accretive operator, then T † is m-accretive. FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 5
Proof.
By assumption,
Re < T x, x > ≥ x ∈ D ( T ) ∩ N ( T ) ⊥ = R ( T † ) . Hence
Re < y, T † y > ≥ y ∈ R ( T ) . Now let x ∈ D ( T † ) = R ( T ) ⊕ N ( T ), then x = x + x , with x ∈ R ( T ) and x ∈ N ( T ).Therefore, Re < x, T † x > = Re < x , T † x > ≥ , which implies Re < x, T † x > ≥ x ∈ D ( T † ) . Since T † is closed densely defined, it follows that T † is m-accretive. (cid:3) It well known that by [5, Theorem 2; p 341], T †† = T , this yields to Corollary 3.3. T † is m-accretive operator if and only if T is m-accretive. Corollary 3.4. T is m-accretive operator with closed range if and only if T † is boundedand accretive. Corollary 3.5. If T is m-accretive operator with closed range, then T is an EP (EqualProjections) operator, that is, T † bounded and T T † = T † T on D ( T ) . Proposition 3.6.
Let T an accretive bounded operator. If W ( T ) ⊆ D and W ( T † ) ⊆ D ,then T is unitary on R ( T ) .Proof. Recall that the numerical radius of the bounded operator T is defined by w ( T ) = sup k x k =1 | < T x, x > | . It is known by [10, Theorem 1.3-1] that the numerical radius is equivalent to the usualoperator norm; w ( T ) ≤ k T k ≤ w ( T ) . Hence, the assumption that w ( T † ) ≤ T † is bounded. Thus R ( T ) is closed.Since T is m-accretive, then R ( T ) = R ( T ∗ ) = N ( T ) ⊥ . We consider the restriction of T from R ( T ) into itself. Since T †|R ( T ) = ( T |R ( T ) ) − , w (( T R ( T ) ) − ) = w ( T †|R ( T ) ) = w ( T † ) ≤ w ( T |R ( T ) ) = w ( T ) ≤ T |R ( T ) ) − and T |R ( T ) , we conclude that T |R ( T ) is unitary on R ( T ). (cid:3) Theorem 3.7.
Let T is m-accretive operator and S is bounded and accretive. We have (1) T + S is m-accretive. (2) If R ( T ) is closed, R ( S ) ⊆ R ( T ) and (cid:13)(cid:13) T † S (cid:13)(cid:13) < . Then • R ( T + S ) = R ( T ) is closed and N ( T + S ) = N ( T ) . • T + S is an EP operator, and ( T + S ) † = ( I + T † S ) − T † = T † ( I + ST † ) − . In particular, T † = ( T + S ) † ( I + ST † ) , and (cid:13)(cid:13) ( T + S ) † − T † (cid:13)(cid:13) ≤ k S k (cid:13)(cid:13) T † (cid:13)(cid:13) − k T † S k . F. BOUCHELAGHEM, M. BENHARRAT
Proof. (1) Clearly, the operator T + S , with D ( T + S ) = D ( T ), is densely defined, closedand accretive. Since also its adjoint operator ( T + S ) ∗ = T ∗ + S ∗ is accretive, the operator T + S is m-accretive.(2) If R ( S ) ⊆ R ( T ), then it is obvious that R ( T + S ) ⊆ R ( T ) and T T † S = P R ( T ) S = S .Conversely, let y ∈ R ( T ), so y = T x for some x ∈ D ( T ). The condition (cid:13)(cid:13) T † S (cid:13)(cid:13) < I + T † S ) − exists and bounded. Hence, there exists a u ∈ D ( T ) such that x = ( I + T † S ) u . This shows that y = T ( I + T † S ) u = T u + Su ∈ R ( T + S ). Hence R ( T ) ⊆ R ( T + S ). Consequently, R ( T + S ) = R ( T ) is closed.Since T and T + S are m-accretive with closed ranges, then N ( T + S ) = R ( T + S ) ⊥ = R ( T ) ⊥ = N ( T ) . Now we prove that ( T + S ) † = ( I + T † S ) − T † . Since, R ( T + S ) is closed and N ( T + S ) = R ( T + S ) , by Corollary 3.5, it follows that T + S is an EP operator.Put T = ( I + T † S ) − T † . We show that T satisfies all the axioms of the Definition 3.1.First let us remark that, since ( I + T † S ) − is invertible, D ( T ) = D ( T † ) = R ( T ) ⊕ N ( T ), N ( T ) = N ( T † ) = R ( T ) ⊥ = N ( T + S ).Let v ∈ R ( T ), then there exists u ∈ D ( T ) such that v = T u = ( I + T † S ) − T † u . Hence T † u = v + T † Sv ∈ R ( T ) ∩ D ( T ). So v = T † u − T † Sv ∈ D ( T ).Now for v ∈ D ( T ), T ( T + S ) v = ( I + T † S ) − T † ( T + S ) v = ( I + T † S ) − T † ( T + T T † S ) v (since S = T T † S )= ( I + T † S ) − T † T ( I + T † S ) v = ( I + T † S ) − P N ( T ) ⊥ ( I + T † S ) v = ( I + T † S ) − P R ( T ) ( I + T † S ) v = ( I + T † S ) − ( I + T † S ) v = v = P R ( T ) v = P R ( T + S ) v = P N ( T + S ) ⊥ v. and for u ∈ D ( T ),( T + S ) T u =( T + S )( I + T † S ) − T † u =( T + T T † S )( I + T † S ) − T † u (since S = T T † S )= T ( I + T † S )( I + T † S ) − T † u = T T † u = P R ( T ) u = P R ( T + S ) u. The uniqueness of ( T + S ) † follows from Definition 3.1.Since R ( S ) ⊆ R ( T ), by Neumann series, we have( I + T † S ) − T † = ∞ X n =0 ( − T † S ) n T † = ∞ X n =0 T † ( − ST † ) n = T † ( I + ST † ) − . (3.1) FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 7
For the last inequality, we can see that( T + S ) † − T † = ( I + ST † ) − T † − ( I + ST † )( I + ST † ) − T † = [ I − ( I + ST † )]( I + ST † ) − T † = ( − ST † )( I + ST † ) − T † . Hence we get the desired inequality. (cid:3)
Remark . Recall that the reduced minimum modulus of a non-zero operator T is definedby γ ( T ) = inf {k T x k : x ∈ N ( T ) ⊥ ∩ D ( T ) , k x k = 1 } . If T = 0 then we take γ ( T ) = ∞ . Note that (see [14]), R ( T ) is closed if an only if γ ( T ) >
0. In that case, γ ( T ) = 1 k T † k , where T † is the Moore-Penrose inverse of T . Letus remark that if we assume that k S k < γ ( T ) instead the condition (cid:13)(cid:13) T † S (cid:13)(cid:13) <
1, then theTheorem 3.7 hold true.
Proposition 3.9.
Assume that T be accretive and T be m-accretive. If R ( T ) is closed,then R ( T ) is closed and γ ( T ) ≥ γ ( T ) .Proof. The operator T is m-( π/ T . By theLandau-Kolmogorov inequality, [16, Theorem.], applied to T , we have k T x k ≤ (cid:13)(cid:13) T x (cid:13)(cid:13) k x k , for all x ∈ N ( T ) ⊥ ∩ D ( T ). It follows that k x k γ ( T ) ≤ k T x k ≤ (cid:13)(cid:13) T x (cid:13)(cid:13) k x k , and hence (cid:13)(cid:13) T x (cid:13)(cid:13) ≥ γ ( T ) k x k , for all x ∈ N ( T ) ⊥ ∩D ( T ). Now by the definition of γ ( T ) we obtain γ ( T ) ≥ γ ( T ) (cid:3) A canonical factorization of a monic quadratic operator pencils
In this section we will investigate a canonical factorization of quadratic operator pencils Q of the form Q ( λ ) = λ I − λB − C, (4.1)on a Hilbert space with domain D ( Q ) = D ( B ) ∩ D ( C ), where λ ∈ C is the spectralparameter and the two operators B and C with domain D ( C ) and D ( B ), respectively,satisfy one of the following conditions,( C.1 ) there exists α ≥
0, 0 ≤ β < δ ≥ Re < B x, Cx > ≥ − α k x k − β (cid:13)(cid:13) B x (cid:13)(cid:13) − δ (cid:13)(cid:13) B x (cid:13)(cid:13) k x k , for all x ∈ D ( B ) ⊂ D ( C ). F. BOUCHELAGHEM, M. BENHARRAT ( C.2 ) C is B -bounded with lower bound <
1. i.e. there exists a ≥ ≤ b < k Cx k ≤ a k x k + b (cid:13)(cid:13) B x (cid:13)(cid:13) , for all x ∈ D ( B ) ⊂ D ( C ) . ( C.3 ) I + C ( B + t ) − is boundedly invertible, for some t > C.4 ) B is accretive and D ( B ) ⊂ D ( C ).( C.5 ) B is accretive and C is bounded. Proposition 4.1.
Let B be m-accretive and C is accretive. If the operator B and C verifies one of the conditions above, then the operator Λ = B + C with domain D ( B ) ism-accretive.Proof. Assume (
C.1 ), then by [21, Theorem 3.10] we can prove that B + C is m-accretive.For the convince of the reader we give a detailed proof of this fact and adapted to theHilbert case. First, we have B + C is accretive densely defined. We show that B + C isclosed. In fact, it follows from ( C.1 ) that (cid:13)(cid:13) B x (cid:13)(cid:13) = Re < B x, B x > ≤ Re < ( B + C ) x, B x > + α k x k + β (cid:13)(cid:13) B x (cid:13)(cid:13) + δ (cid:13)(cid:13) B x (cid:13)(cid:13) k x k , for all x ∈ D ( B ). So, we have(1 − β ) (cid:13)(cid:13) B x (cid:13)(cid:13) ≤ [ δ k x k + (cid:13)(cid:13) ( B + C ) x (cid:13)(cid:13) ] (cid:13)(cid:13) B x (cid:13)(cid:13) + α k x k , for all x ∈ D ( B ). Solving this inequality, we obtain (cid:13)(cid:13) B x (cid:13)(cid:13) ≤ − β (cid:13)(cid:13) ( B + C ) x (cid:13)(cid:13) + κ k x k (4.2)for all x ∈ D ( B ), with κ = α + p δ (1 − β )1 − β . On the other hand, since D ( B ) ⊂ D ( C ),with D ( B ) dense in H , there exists a constant ϑ >
0, such that k Cx k ≤ ϑ ( k x k + (cid:13)(cid:13) B x (cid:13)(cid:13) ) , (4.3)for all x ∈ D ( B ). Now, let a sequence ( x n ) n ⊂ D ( B ) such that x n −→ x and ( B + C ) x n −→ y . Applying the inequality (4.2) to x replaced by x n − x m , we see that thesequence ( B x n ) n converge. Since B is closed we conclude that B x n −→ B x and x ∈ D ( B ). By (4.3), we have k C ( x n − x ) k ≤ ϑ k x n − x k + (1 + ϑ ) (cid:13)(cid:13) B ( x n − x ) (cid:13)(cid:13) . Hence ( B + C ) x n −→ ( B + C ) x and y = ( B + C ) x , which shows B + C is closed. Onthe other hand, we have Re < B x, tCx > ≥ − tα k x k − tβ (cid:13)(cid:13) B x (cid:13)(cid:13) − tδ (cid:13)(cid:13) B x (cid:13)(cid:13) k x k , for all 0 ≤ t ≤
1. Since tβ <
1, by the same argument, we assert that ( B + tC ) is closedfor all 0 ≤ t ≤
1. Hence, ( B + tC ) is closed and accretive for all 0 ≤ t ≤
1. By [21,Lemma 3.1], the dimension of R ( B + C + λ ) ⊥ and R ( B + λ ) ⊥ are the same for all λ > B is m-accretive, we conclude that B + C is m-accretive. FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 9
If (
C.2 ) holds, the result follows by [9, Theorem 2.]. Also (
C.2 ) implies (
C.1 ) in thecase of γ = 0, see [20, Remark 4.4]. In fact, setting α = a/ β = ( b + 1) /
2, we havethat for x ∈ D ( B ), k Cx k ≤ α k x k + (2 β − (cid:13)(cid:13) B x (cid:13)(cid:13) ≤ (cid:13)(cid:13) ( B + C ) x (cid:13)(cid:13) − (cid:13)(cid:13) B x (cid:13)(cid:13) + 2 α k x k + 2 β (cid:13)(cid:13) B x (cid:13)(cid:13) = 2( Re < ( B + C ) x, x > + α k x k + β (cid:13)(cid:13) B x (cid:13)(cid:13) ) + k Cx k . Hence
Re < ( B + C ) x, x > + α k x k + β k B x k ≥ C.3 ) is satisfied. Since B + C is densely defined and accretive, it sufficesto show that R ( B + C + t ) = H . But this follows immediately from B + C + t = ( I + C ( B + t ) − )( B + t ) , and clearly B + C + t is invertible.Now, we consider ( C.4 ). Since B is an accretive operator, by [11, Theorem 1.2], wehave for an arbitrary ν > k Bx k ≤ ν k x k + 1 ν (cid:13)(cid:13) B x (cid:13)(cid:13) , (4.4)for all x ∈ D ( B ). Since D ( B ) ⊂ D ( C ), with D ( B ) dense in H , there exists a constant η >
0, such that k Cx k ≤ η k Bx k , for all x ∈ D ( B ). It follows that k Cx k ≤ η ( ν k x k + 1 ν (cid:13)(cid:13) B x (cid:13)(cid:13) ) , for all x ∈ D ( B ). Choosing ν > ην <
1, we get C is B -bounded withlower bound < C.5 ) is a particular case of (
C.4 ). (cid:3) Remark . (1) In ( C.3 ), if we assume further D ( B ) ⊂ D ( C ), then by [25, Propo-sition 2.12], the lower bound b in ( C.2 ) is equal to sup t> k C ( B + t ) − k . Hence,if we assume further, k C ( B + t ) − k < t >
0, so I + C ( B + t ) − isboundedly invertible, for some t >
0. In this case (
C.3 ) implies (
C.2 ).(2) If the condition (
C.4 ) is satisfied, clearly D ( Q ) = D ( B ). In the sequel, we assume that the operator B and C verifies one of the con-ditions of the Proposition 4.1 with D ( B ) ⊂ D ( C ) , unless otherwise specified. Now, we state some properties of the operator Λ = B + C . The first important one isthe existence and uniqueness of its square root. This is an immediate consequence of [14,Theorem 3.35, p. 281]. Corollary 4.3.
The operator Λ admits unique square root Λ m- ( π/ -accretive operatorwith D ( B ) is a core of Λ (that is, the closure of the restriction of Λ to D ( B ) is again Λ ). Proposition 4.4. If C is θ -accretive, with ≤ θ < π/ , then N (Λ) ⊂ N ( B ) ∩ N ( C ∗ ) . Proof. (1) Let x ∈ D ( B ), x = 0, such that Λ x = 0, as before, we have Re < Λ x, x > = Re < B x, x > + Re < Cx, x >, then
Re < B x, x > ≤ Re < Λ x, x > and Re < Cx, x > ≤ Re < Λ x, x > . Therefore,
Re < Λ x, x > = 0 implies that Re < B x, x > = 0 and Re < Cx, x > = 0. Onthe other hand, since C is θ -accretive, with 0 ≤ θ < π/
2, then | Im < Cx, x > | ≤ tan ( θ ) Re < Cx, x > .
Thus,
Im < Cx, x > = 0 and
Im < B x, x > = − Im < Cx, x > = 0 , hence < B x, x > = 0 and < Cx, x > = 0 . Since B is m-accretive operator, we conclude that x ∈ N ( B ) and x ∈ N ( C ∗ ). (cid:3) By the same way as in [21, Lemma 1.8], we obtain,
Proposition 4.5.
Let B be m-accretive and C is accretive such that the condition ( C.1 )holds. If R ( B ) is closed and δγ ( B ) − + αγ ( B ) − + β < , then R (Λ) is closed andhence the Moore-Penrose inverse of Λ is bounded. In particular, if B is injective, then Λ boundedly invertible.Proof. Let x ∈ N ( B ) ⊥ ∩ D ( B ). Since (cid:13)(cid:13) B x (cid:13)(cid:13) ≥ γ ( B ) k x k , it follows from ( C.1 ), that
Re < B x, Cx > ≥ − ( δγ ( B ) − + αγ ( B ) − + β ) (cid:13)(cid:13) B x (cid:13)(cid:13) . Hence
Re < B x, ( C + B ) x > ≥ [1 − ( δγ ( B ) − + αγ ( B ) − + β )] (cid:13)(cid:13) B x (cid:13)(cid:13) . Therefore, (cid:13)(cid:13) ( C + B ) x (cid:13)(cid:13) ≥ [1 − ( δγ ( B ) − + αγ ( B ) − + β )] (cid:13)(cid:13) B x (cid:13)(cid:13) ≥ υ k x k , with υ = γ ( B )[1 − ( δγ ( B ) − + αγ ( B ) − + β )] >
0. So the desired result. (cid:3)
Combining Propositions 3.9 and 4.5, we obtain
Corollary 4.6.
Let B be m-accretive and C be accretive such that the condition ( C.1 )holds. If R ( B ) is closed and δγ ( B ) − + 2 αγ ( B ) − + β < , then R (Λ) is closed andhence the Moore-Penrose inverse of Λ is bounded. In particular, if B is injective, then Λ boundedly invertible. By Theorem 3.7, we have
Corollary 4.7.
Let B is m-accretive with closed range and C is accretive such that thecondition ( C.5 ) holds. If R ( C ) ⊆ R ( B ) and (cid:13)(cid:13) ( B ) † C (cid:13)(cid:13) < Then • R (Λ) = R ( B ) is closed and N (Λ) = N ( B ) . FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 11 • Λ is an EP operator, and Λ † = ( I + ( B ) † C ) − ( B ) † = ( B ) † ( I + C ( B ) † ) − . Remark . By Proposition 3.9, the Corollary 4.7 hold true if we assume only R ( B ) isclosed.Now, we define the linear factors Z and Z into which the quadratic pencil can bedecomposed. Lemma 4.9.
Assume that B is accretive. The operators Z = B + Λ and Z = B − Λ with domain D ( B ) ∩ D (Λ ) are B -bounded with lower bound < and closable operators.Further, the closure of Z is m- ( π/ -accretive operator and if B m- ( π/ -accretive, then N ( Z ) = N ( B ) ∩ N (Λ ) .Proof. Since D ( B ) is dense in H , so is D ( C ). Consequently, as before, there exist non-negative constant α such that k Cx k ≤ α ( k x k + (cid:13)(cid:13) B x (cid:13)(cid:13) ) , (4.5)for all x ∈ D ( B ). On the other hand, by [19, Theorem 6.10], we have for an arbitrary ρ > (cid:13)(cid:13)(cid:13) Λ x (cid:13)(cid:13)(cid:13) ≤ π ( ρ k x k + 1 ρ k Λ x k ) , (4.6)for all x ∈ D ( B ). By (4.4), (4.5) and (4.6), it follows that k Z i x k ≤ k Bx k + 2 (cid:13)(cid:13)(cid:13) Λ x (cid:13)(cid:13)(cid:13) ≤ ν k x k + 1 ν (cid:13)(cid:13) B x (cid:13)(cid:13) ) + 2 π ( ρ k x k + 1 ρ k Λ x k ) ≤ ν k x k + 1 ν (cid:13)(cid:13) B x (cid:13)(cid:13) ) + 2 ρπ k x k + 4 π ρ ( (cid:13)(cid:13) B x (cid:13)(cid:13) + k Cx k ) ≤ (2 ν + ρπ + 2 α ρπ ) k x k + 2 π ( 1 ν + 2( α + 1) ρ ) (cid:13)(cid:13) B x (cid:13)(cid:13) ≤ α k x k + β (cid:13)(cid:13) B x (cid:13)(cid:13) , for some α , β > i = 1 , x ∈ D ( B ). Since ν and ρ are arbitrary, we canchoose β <
1. In addition, both of Z and Z are densely defined on H with numericalrange is not the whole complex plane, it follows that Z and Z are closable operators.Since Z is ( π/ Z ∗ is ( π/ π/ N ( B ) ∩ N (Λ ) ⊂ N ( Z ) is obvious. Conversely, let x ∈ D ( Z ), x = 0,such that Z x = 0, we have Re < Z x, x > = Re < Bx, x > + Re < Λ x, x >, then Re < Bx, x > ≤ Re < Z x, x > and Re < Λ x, x > ≤ Re < Z x, x > . Therefore,
Re < Z x, x > = 0 implies that Re < Bx, x > = 0 and
Re < Λ x, x > = 0. Onthe other hand, since B and Λ are ( π/ | Im < Bx, x > | ≤
Re < Bx, x > and (cid:12)(cid:12)(cid:12)
Im < Λ x, x > (cid:12)(cid:12)(cid:12) ≤ Re < Λ x, x > . Thus,
Im < Bx, x > = 0 and
Im < Λ x, x > = 0 , hence < Bx, x > = 0 and < Λ x, x > = 0 . Since B and Λ are m-( π/ Bx = 0 and Λ x = 0 . Consequently, N ( Z ) ⊂ N ( B ) ∩ N (Λ ) . (cid:3) Now, since D ( B ) is a core of both B and Λ , we have Corollary 4.10.
The closure of the restriction of Z i to D ( B ) is again Z i , i = 1 , . Now we are in position to give a factorization of the quadratic operator pencil (4.1).
Proposition 4.11.
Assume that Z ( D ( B )) ⊂ D ( Z ) . Q admits the following canonicalfactorization, Q ( λ ) x = 12 ( λI − Z )( λI − Z ) x + 12 ( λI − Z )( λI − Z ) x, (4.7) for all x ∈ D ( B ) .In particular, if B Λ = Λ B on D ( B ) , then Q ( λ ) x = ( λI − Z )( λI − Z ) x = ( λI − Z )( λI − Z ) x, (4.8) for all x ∈ D ( B ) .Proof. We can easily verify that Z x − BZ x − Z Bx − Cx = 0 , for all x ∈ D ( B ), hence on D ( B ), we have Q ( λ ) = Q ( λ ) − ( Z − BZ − Z B − C )= λ I − λB − C − Z + BZ + Z B + C = λ I − Z − B ( λ − Z ) − ( λ − Z ) B = 12 ( λ − Z )( λI + Z − B ) + 12 ( λI + Z − B )( λ − Z )= 12 ( λI − Z )( λI − Z ) + 12 ( λI − Z )( λI − Z ) . Now, if if B Λ = Λ B on D ( B ), we obtain (4.8). (cid:3) As seen before the closure of Z is m-( π/ Z = B + Λ is closed? FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 13
Proposition 4.12.
Assume further, B closed and accretive with D (Λ ) ⊂ D (Λ α ) ⊂D ( B ) , for some < α < / , then Z and − Z are m- ( π/ -accretive operators. In par-ticular, − Z and Z generates holomorphic C -semigroup of contraction operators T ( z ) and T ( z ) of angle π , respectively.Proof. If D (Λ ) ⊂ D (Λ α ) ⊂ D ( B ), for some 0 < α < /
2, it follows that, by [25, Corollary2.14], B + Λ / is closed and − ( B + Λ / ) generates a strongly continuous semigroup ofcontraction operators. This implies that Z is m- π/ t >
0, we have( t + Z ) x = ( t + B + Λ ) x = [ I + B ( t + Λ ) − ]( t + Λ ) x, for all x ∈ D ( Z ). Since ( t + Z ) and t + Λ are invertible, then I + B ( t + Λ ) − is alsoinvertible. Also, (cid:13)(cid:13)(cid:13) B ( t + Λ ) − (cid:13)(cid:13)(cid:13) < , for all t > . In fact, for any t > (cid:13)(cid:13)(cid:13) B ( t + Λ ) − (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) B ( I + Λ ) − α ( I + Λ ) α ( t + Λ ) − (cid:13)(cid:13)(cid:13) . By assumption, the closed graph theorem yields B ( I + Λ ) − α is bounded. On the otherhand, (cid:13)(cid:13)(cid:13) ( I + Λ )( t + Λ ) − (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) (( I + Λ )( t + Λ ) − ) α (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ( t + Λ ) α − (cid:13)(cid:13)(cid:13) ≤ Kt α − , since ( I + Λ )( t + Λ ) − is uniformly bounded for t >
1. Hence (cid:13)(cid:13)(cid:13) B ( t + Λ ) − (cid:13)(cid:13)(cid:13) −→ t −→ + ∞ .Now, we have( − t + Z ) x = ( − t + B − Λ ) x = [ B ( t + Λ ) − − I ]( t + Λ ) x, for all x ∈ D ( Z ). Since the operators on the right-hand side are invertible for all t > Z − t ) is invertible for all t >
0. This implies that ( −∞ , ⊂ ρ ( − Z ). Since Λ ism-( π/ ε > (cid:13)(cid:13)(cid:13) ( λ + Λ ) − (cid:13)(cid:13)(cid:13) ≤ M ε | λ | , for | arg ( λ ) | ≤ π π − ε with M ε is independent of λ (see [14, pp. 490]). Therefore, ( Z − λ ) is invertible and (cid:13)(cid:13) ( Z − λ ) − (cid:13)(cid:13) ≤ M ε | λ | (1 − M ) , for | arg ( λ ) | ≤ π π − ε, for all ε >
0. This implies that ρ ( − Z ) contains also the sector | arg ( λ ) | < π π − Z is m- π − Z and Z generates holomorphic C -semigroup T ( z ) and T ( z ) of angle π (cid:3) Remark . In Proposition 4.1, if we assume only B is m-accretive, C + B need not bem-accretive, because B fails to be accretive (with the same vertex as B ) even in the caseof an accretive matrix B with numerical range contained in a sector of angle less than π/
4, as the following example shows.
Example 4.14.
Let H = C and B = (cid:20) − i i i
16 + 4 i (cid:21) . For x = ( x , x ) ∈ C ; we have Re < Bx, x > = 4 | x | + 16 | x | and Im < Bx, x > = − | x | + 8 Re ( x x ) + 4 | x | ≤ | x | + 8 | x | < Re < Bx, x > . Thus W ( B ) ( S π/ . However, for x = (1 , < B x, x > = − − i, it follows that W ( B ) is not a subset of the right half complex plane. Remark . The operator pencil Q is not necessarily an accretive, because we can findan eigenvalues not located in the closed right half-plane. Indeed, let λ be an eigenvalueof Q and v ∈ D ( Q ) its corresponding eigenvector with k v k = 1. Let us remark that if λ = 0, then 0 = < Cv, v > and hence 0 ∈ W ( C ). In the sequel we assume that λ = 0 with λ = α + iβ . Then < Q ( λ ) v, v > = 0 , and consequently, taking real and imaginary parts,( α − β ) − αRe < Bv, v > +2 βIm < Bv, v > − Re < Cv, v > = 0 , and 2 αβ − βRe < Bv, v > − αIm < Bv, v > − Im < Cv, v > = 0 . It follows that
Re < Cv, v > = ( α − β ) − αRe < Bv, v > +2 βIm < Bv, v >, and Im < Cv, v > = 2 αβ − βRe < Bv, v > − αIm < Bv, v > . Since
Re < Cv, v > ≥
0, we obtain from the first relation,2 αRe < Bv, v > ≤ α − β + 2 βIm < Bv, v > . The fact that | Im < Bv, v > | ≤
Re < Bv, v > , we get2 αRe < Bv, v > ≤ α − β + 2 | β | Re < Bv, v > .
Thus 2( α − | β | ) Re < Bv, v > ≤ α − β . FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 15
Now, if assume | α | ≤ | β | , it follows that( α − | β | ) Re < Bv, v > ≤ . Consequently, α ≤ | β | .5. An application to an abstract second order differential equation
Let us consider, in the complex Hilbert space H , the following abstract second orderdifferential equation u ′′ ( x ) − Bu ′ ( x ) − Cu ( x ) = f ( x ) , x ∈ (0 , , (5.1)under the boundary conditions u (0) = u , u (1) = u , (5.2)where B and C are two closed operators in a Hilbert space with domains D ( B ) and D ( C ),respectively, f ∈ L p (0 , H ), 1 < p < ∞ and u , u are given elements in H . We seek fora strict solution u to (5.1)-(5.2), i.e. a function u such that (cid:26) i ) u ∈ W ,p (0 , H ) ∩ L p (0 , D ( C ))) , u ′ ∈ L p (0 , D ( B )) ,ii ) u satisfies (5.1) − (5.2) . (5.3) Theorem 5.1.
Let B and C two operators in a Hilbert space H such that (1) B is m-accretive and C is accretive satisfy one of conditions of Proposition 4.1. (2) B is closed and accretive with D (( B + C ) ) ⊂ D (( B + C ) α ) ⊂ D ( B ) , for some < α < / . (3) B ( D (( B + C ) )) ⊂ D (( B + C ) ) . (4) ( B + C ) − exist and bounded. (5) B ( B + C ) = ( B + C ) B on D ( B ) . (6) f ∈ L p (0 , H ) with < p < ∞ .Then the problem (5.1) - (5.2) has a classical solution u if and only if Z e . − Z u , Z e . − Z u ∈ L p (0 , H ) . In this case, u is uniquely determined by u ( x ) = ( I − e Z − Z ) − e xZ u + ( I − e Z − Z ) − e − (1 − x ) Z u − ( I − e Z − Z ) − e xZ e − Z (cid:18) u − ( Z − Z ) − Z e (1 − s ) Z f ( s ) ds (cid:19) − ( I − e Z − Z ) − e − (1 − x ) Z e Z (cid:18) u − ( Z − Z ) − Z e − sZ f ( s ) ds (cid:19) − ( I − e Z − Z ) − ( Z − Z ) − e xZ Z e − sZ f ( s ) ds + ( I − e Z − Z ) − ( Z − Z ) − e − (1 − x ) Z Z e − (1 − s ) Z f ( s ) ds + ( Z − Z ) − Z x e ( x − s ) Z f ( s ) ds + ( Z − Z ) − Z x e ( x − s ) Z f ( s ) ds. Proof.
Under the assumptions, by Proposition 4.12, the factors − Z and Z generatesbounded holomorphic C -semigroup ( e − tZ ) t ≥ and ( e tZ ) t ≥ , respectively. Also, D ( Z ) = D ( Z ) = D (Λ / ) and D ( Z Z ) = { x ∈ D ( Z ); Z x ∈ D ( Z ) } = { x ∈ D ( Z ); Z x ∈ D ( Z ) } = D ( Z ) , D ( Z Z ) = { x ∈ D ( Z ); Z x ∈ D ( Z ) } = { x ∈ D ( Z ); Z x ∈ D ( Z ) } = D ( Z ) . But D ( Z ) = (cid:8) x ∈ D (Λ / ); Z x ∈ D (Λ / ) (cid:9) and D ( Z ) = (cid:8) x ∈ D (Λ / ); Z x ∈ D (Λ / ) (cid:9) . The fact that, B ( D (( B + C ) )) ⊂ D (( B + C ) ), we obtain D ( Z ) = D ( Z ). Furthermore, e tZ u ∈ D ( Z n ) and e − tZ u ∈ D ( Z n ) for all u , u ∈ H , t > n ∈ N . Hence u ( x ) ∈ D ( C ) for all x ∈ (0 , C -semigroups are holomorphic, u ( . ) can bedifferentiated any numbers of times. Now, by taking − B instead B , A = − C , L = − Z and M = Z in [8, Theorem 5.], all assumptions of this theorem are fulfilled. Hence weobtain the desired result. (cid:3) An example of a second order partial differential equation
The aim of this section is to use the obtained results to discuss the existence, uniqueness,and maximal regularity of the strict solution for the following non-homogeneous secondorder differential equation,( E ) ∂ u∂x ( x, y ) − p ( y ) ∂ u∂y∂x ( x, y ) − p ( y ) ∂u∂x ( x, y ) + αp ( y ) ∂u∂y ( x, y )+( αp ( y ) + β ) u ( x, y ) − γu ( x, y ) = f ( x, y ) , x ∈ (0 , , y ∈ (0 , u (0 , y ) = u ( y ) , u (1 , y ) = u ( y ) , y ∈ (0 , u ( x,
0) = u ( x,
1) = 0 , x ∈ (0 , ∂u∂x ( x,
0) = ∂u∂x ( x,
1) = 0 , x ∈ (0 , • f ∈ L p (0 , L (0 , C )), 1 < p < ∞ , • α ∈ R , β ∈ C , p , p ∈ C (0 ,
1) and p = 0. • γ = − ( r + 14 ε M + M ), with r > ε are arbitrary and chosen such that m − ε (1 + r ) M >
0, for some nonegative constants m , M and M are describedbelow.The second order differential equation ( E ) is equivalent to ∂ u∂x ( x, y ) − B ∂u∂x ( x, y ) + Cu ( x, y ) − γu ( x, y ) = f ( x, y ) , x ∈ (0 , , y ∈ (0 , . (6.1)with the boundary conditions u (0 , y ) = u ( y ) , u (1 , y ) = u ( y ) , y ∈ (0 , , (6.2) FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 17 where, B = p ∂∂y + p , D ( B ) = { ψ ∈ H (0 ,
1) : ψ (0) = ψ (1) = 0 } and C = αp ∂∂y + ( αp + β ) , D ( C ) = { φ ∈ H (0 ,
1) : φ (0) = φ (1) = 0 } . with φ ( y ) = u ( x, y ) and ψ ( y ) = ∂u∂x ( x, y ), x ∈ (0 , , y ∈ (0 , u ( ., y ) to (6.1)-(6.2), i.e. a function u ( ., y ) ∈ L (0 , C ) such that 5.3 holds. Thiswill be done by the following preparatory results. Claim 1.
The operator − B − γI is m- ω -accretive, with ω = arctan( 1 r ) .Proof. For ψ ∈ D ( B ) = { ψ ∈ H (0 ,
1) : ψ (0) = ψ (1) = 0 } ⊂ D ( B ), we have − B ψ = ϕ ψ ′′ + ϕ ψ ′ + ϕ ψ, with ϕ = − p , ϕ = − p ( p ′ + 2 p ) and ϕ = − ( p + p p ′ ) . Under the assumptions thereexists a nonegative constants m , M and M such that − ϕ > m > , | ϕ − ϕ ′ | ≤ M , and | ϕ | ≤ M . (6.3)By [14, Example V-3.34], − B is m- ω -accretive operator with vertex γ , where γ = − ( r + 14 ε M + M ), ω = arctan( 1 r ), r > ε is chosen such that m − ε (1 + r ) M > − B − γI is m- ω -accretive, with ω = arctan( 1 r ). (cid:3) Claim 2. If αp + Re ( β ) − α p ′ ≥ then C is an accretive operator.Proof. Let ψ ∈ D ( C ), we have < Cψ, ψ > = α Z p ( y ) < ψ ′ ( y ) , ψ ( y ) > dy + Z ( αp ( y ) + β ) | ψ ( y ) | dy. By integration by parts, < Cψ, ψ > = − α Z p ( y ) < ψ ( y ) , ψ ′ ( y ) > dy + Z ( αp ( y ) + β − αp ′ ( y )) | ψ ( y ) | dy. Also < Cψ, ψ > = < ψ, Cψ > = α Z p ( y ) < ψ ( y ) , ψ ′ ( y ) > dy + Z ( αp ( y ) + β ) | ψ ( y ) | dy. Thus
Re < Cψ, ψ > = Z ( αp + Re ( β ) − α p ′ ) | ψ ( y ) | dy. Hence the desired result. (cid:3)
Claim 3. If αp + Re ( β ) − α p ′ ≥ , then − Λ = − B + C − γI with domain D ( B ) is m-accretive. Also, − Λ admits an unique square root ( − Λ) / m- ( π/ -accretive. Proof.
By Claim 1. − B − γI with domain D ( B ) is m-accretive and C is an accretiveby Claim 2. Also, D ( B ) = D ( C ). Now the desired result holds from the third item ofProposition 4.1. (cid:3) Claim 4. If p ′′ is continuous on [0 , , then ( − B + C − γI ) − exist and bounded.Proof. As before; for ψ ∈ D ( B ) = { ψ ∈ H (0 ,
1) : ψ (0) = ψ (1) = 0 } ⊂ D ( B ), we have[ − B + C − γI ] ψ = ϕ ψ ′′ + ( ϕ + αp ) ψ ′ + ( ϕ + αp + β − γ ) ψ, with ϕ = − p , ϕ = − p ( p ′ + 2 p ) and ϕ = − ( p + p p ′ ) . Since p ′′ and p ′ are continuouson [0 , ϕ ′′ , ϕ ′ + αp ′ and ϕ + αp + β − γ are are continuous on [0 , − B + C − γI ) − exist and bounded. (cid:3) Combining Claim 3., Lemma 4.9 and corollaries 4.3 and 4.10, we obtain,
Claim 5.
The operators Z = B − ( − Λ) / and Z = B + ( − Λ) / with domain D ( B ) ∩ D (Λ ) are B -bounded and closable operators. Furthermore, the closure of therestriction of Z i to D ( B ) is again Z i , i = 1 , . We are now ready to state the following existence and uniqueness result.
Theorem 6.1.
Let the equation ( E ) on H = L (0 , C ) . Assume that (1) f ∈ L p (0 , H ) , < p < ∞ , (2) α ∈ R , β ∈ C , p ∈ C (0 , , p ∈ C (0 , and p = 0 , (3) p − p ′ ≥ and α ( p − p ′ ) + Re ( β ) ≥ , (4) γ = − ( r + 14 ε M + M ) , with r > and ε are arbitrary and chosen such that m − ε (1 + r ) M > , for some nonegative constants m , M and M are given by (6.3) .Then the problem (6.1) - (6.2) has a classical solution u if and only if Z e . − Z u , Z e . − Z u ∈ L p (0 , H ) . In this case, u is uniquely determined as in Theorem 5.1.Proof. Let us remark first that p − p ′ ≥ B is an accretive operator. Itsuffices to take α = 1 and β = 0 in Claim 2. Thus the restriction of Z and − Z to D ( B )are m-( π/ B ( − Λ) / = ( − Λ) / B on D ( B ).By Claim 4., the inverse of ( − Λ) / exist and bounded. Thus, all assumptions of Theorem5.1 are fulfilled. Consequently, we get the desired result. (cid:3) Open problems
In this section, we summarize some of the open problems discussed so far.(1) In the case of selfadjoint quadratic pencil, the properties of the operators Z and Z in the factorization (4.8) are well known under the assumption that the spectralzones are separated [13, 17]: these operators are similar to self adjoint operators.In particular, if the spectrum of the pencil Q ( . ) in one of these zones is discrete,then the eigenvectors corresponding to the eigenvalues of the pencil in this zone FACTORIZATION OF QUADRATIC PENCILS OF ACCRETIVE OPERATORS 19 form a basis equivalent to an orthonormal basis, or a Riesz basis. In the caseof adjoining spectral zones the properties of Z and Z have not been studiedextensively. Some answers for the selfadjoint operators case are given in [23], butin general the question is steel open.(2) In view of the Proposition 4.12, under what conditions the factors Z and Z verified the statements of this proposition?(3) In [6] Duffin proved a variational principle for eigenvalues of a quadratic matrixpolynomial, which was generalized in various directions to more general operatorfunctions. In [7] such a variational principle was proved for eigenvalues of op-erator functions whose values are possibly unbounded self-adjoint operators. aninteresting problem is to adapt this variational principle from [7] to our situation.(4) It well known that the linearized operator associate to Q ( . ), is given by A = (cid:20) IC B (cid:21) in appropriate domain. Can use this approach of factoring Q ( . ) to studying thespectral properties of A and conversely? References [1] S. D. Aglazin, I. A. Kiiko,
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E-mail address : [email protected] D´epartement de G´enie des syst´emes, Ecole Nationale Polytechnique d’Oran-MauriceAudin (Ex. ENSET d’Oran), BP 1523 Oran-El M’naouar, 31000 Oran, Alg´erie.
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