A perturbation result of m-accretive linear operators in Hilbert spaces
aa r X i v : . [ m a t h . F A ] S e p A PERTURBATION RESULT OF M-ACCRETIVE LINEAROPERATORS IN HILBERT SPACES
MOHAMMED BENHARRAT ∗ Abstract.
A new sufficient condition is given for the sum of linear m-accretive operatorand accretive operator one in a Hilbert space to be m-accretive. As an application, anextended result to the operator-norm error bound estimate for the exponential Trotter-Kato product formula is given. introduction A linear operator T with domain D ( T ) in a complex Hilbert space H is said to beaccretive 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 > . Further, if R ( λ + T ) = H for some (and hence for every) λ >
0, we say that T is m-accretive. In particular, every m-accretive operator is accretive and closed densely defined,its adjoint is also m-accretive (cf. [7], p. 279). Furthermore,( λ + T ) − ∈ B ( H ) and (cid:13)(cid:13) ( λ + T ) − (cid:13)(cid:13) ≤ λ for λ > , where, B ( H ) denote the Banach space of all bounded linear operators on H . In particular,a bounded accretive operator is m-accretive.Consider two linear operators T and A in the Hilbert space H , such that D ( T ) ⊂ D ( A ).Assume furthermore that T is m-accretive and A is an accretive operator. Then thequestion is:Under which conditions the sum T + B is m-accretive?Many papers have been devoted to this problem and most results treat pairs T , A ofrelatively bounded or resolvent commuting operators. We refer the reader to [2, 3, 5, 6,15, 17, 18, 20, 21, 22]. Since T is closed it follows that there are two nonnegative constants a , b such that k Ax k ≤ a k x k + b k T x k , for all x ∈ D ( T ) ⊂ D ( A ) . (1.1)In this case, A is called relatively bounded with respect to T or simply T -bounded, andrefer to b as a relative bound. Gustafson [4], generalizing basic work of Rellich, Kato,and others (cf. [7]), showed that that T + A is also m-accretive if A is T -bounded, with Date : 25/07/2020. ∗ Corresponding authorThis work was supported by the Algerian research project: PRFU, no. C00L03ES310120180002.2010
Mathematics Subject Classification.
Primary 47A10; 47A56.
Key words and phrases.
Accretive operators, Perturbation theory, Trotter-Kato product formula. b < T + A ism-accretive, if the bounded operator A ( t + T ) − on H is a contraction for some t >
0, [14,Theorem 1.]. In particular, he also showed that the validity of (1.1) with b = 1 impliesthat the closure of T + A is m-accretive, [14, Corollary 1.]. Later, the same author in [13]gave a variant of perturbation by assumed the existence of two nonnegative constants a and β ≤ Re < T x, Ax > + a k x k + β k T x k ≥ , for all x ∈ D ( T ) . (1.2)If β <
1, then T + A is m-accretive and also the closure of T + A is m-accretive for β = 1,[13, Theorem 4.1]. Note that this result cover the case of relatively bounded perturbation,see [13, Remark 4.4]. There are many papers on the question of such perturbation, see[15, 16, 17, 19, 21] for more results.The aim of this paper is to establish a new perturbation results on the m-accretivity ofthe operator T + A . This may be viewed as a slight improvement and generalization ofthe perturbation results, particularly, those of Okazawa, [15, 13]. The following lemma isour partial answer to the question above. Lemma 1.1.
Let T and A two operators such that D ( T ) ⊂ D ( A ) . Assume that T ism-accretive, A is accretive and there exists c ≥ , such that Re < T x, Ax > ≥ c k Ax k , for all x ∈ D ( T ) . (1.3) If we take b = max { c ≥ holds } , we have, (1) if ≤ b ≤ , then T + A is also m-accretive, (2) if b > then T + A is m- ω -accretive, with ω = π/ − arcsin( b − b ) . Here, T is m- ω -accretive if e ± iθ T is m-accretive for θ = π − ω , 0 < ω ≤ π/
2. In thiscase, − T generates an holomorphic contraction semigroup on the sector | arg ( λ ) | < ω . Inthis connection, we note that for any ε > (cid:13)(cid:13) ( λ + T ) − (cid:13)(cid:13) ≤ M ε | λ | , for | arg ( λ ) | ≤ π ω − ε with M ε is independent of λ (see [7, pp. 490]).The novelty of the lemma is the optimality of b such that (1.3) holds. Clearly, (1.3)implies Re < T x, Ax > ≥ x ∈ D ( T ), this exactly the assumption of [14, Theorem2.]. Hence, we conclude that T + A is also m-accretive. Our result is a refinement of thisresult by given a more precise sector containing the numerical range in function of theconstant b . Also, from (1.3), we have for b > k Ax k ≤ b k T x k , for all x ∈ D ( T ) . (1.4)Thus the assumption (1.3) is stronger than the relative boundedness with respect to T .In particular, if b > b <
1, so according to [4, Theorem 2.], T + A is m-accretive. Here, we say more, T + A is m- ω -accretive with ω depends of the lowerbound 1 b < PERTURBATION RESULT OF M-ACCRETIVE LINEAR OPERATORS 3 Proof of the Lemma
Proof of Lemma 1.1.
Let b = max { c ≥ } . If b = 0, this exactly the [14,Theorem 2.]. Assume that 0 ≤ b ≤
1. We obtain from (1.3)0 ≤ Re < T x, Ax > − b k Ax k ≤ Re < T x, Ax > +( α − b ) k Ax k for some α >
1. Using (1.2), we get0 ≤ Re < T x, Ax > + α − bb k T x k . Choosing α such that β = α − bb <
1, by (1.2) we conclude that T + A is m-accretive(cf.[13, Theorem 4.1]).Now, suppose that that b >
1. Let x ∈ D ( T ), then for every t >
0, we have
Re < tx + T x, Ax > = tRe < x, Ax > + Re < T x, Ax > ≥ b k Ax k . Thus we have k Ax k ≤ b k tx + T x k . (2.1)Since T is m-accretive, then (cid:13)(cid:13) A ( t + T ) − x (cid:13)(cid:13) ≤ b k x k , for all x ∈ H . Hence it follows that (cid:13)(cid:13) A ( t + T ) − (cid:13)(cid:13) ≤ b < . (2.2)Then the operator I + A ( t + T ) − is invertible and (cid:13)(cid:13) ( I + A ( t + T ) − ) − (cid:13)(cid:13) ≤ bb − . The fact that t + T + A = [ I + A ( t + T ) − ]( t + T ) , it follows that − t ∈ ρ ( T + A ) and (cid:13)(cid:13) t ( t + T + A ) − (cid:13)(cid:13) ≤ bb − M, for all t > , with M >
1. Since T + A is accretive, ρ ( T + A ) contains also the half plane { z ∈ C : Re ( z ) < } . Put S = { z ∈ C : z = 0; | arg ( z ) | < π/ − arcsin( 1 M ) = θ } and M ′ := 1 / sin( π/ − θ ′ ) with 0 < θ < θ ′ < π/
2, clearly M ′ > M . Let µ ∈ C suchthat | arg ( µ ) | ≤ θ ′ and fix λ with Reλ = − t <
0. Let | µ − λ | ≤ | λ | M ′ , we have that k ( µ − λ )( t + T + A ) − k ≤ MM ′ < . Hence it follows that µ ∈ ρ ( T + A ) and( µ + T + A ) − = ( λ + T + A ) − [ I + ( µ − λ )( λ + T + A ) − ] − . M. BENHARRAT
Thus (cid:13)(cid:13) µ ( µ + T + A ) − (cid:13)(cid:13) ≤ | µ || λ | − MM ′ M ≤ (1 + 1 M ′ ) 11 − MM ′ M. On the other hand,(1 + 1 M ′ ) 11 − MM ′ M = 1 + sin( π/ − θ ′ )sin( π/ − ω ) − sin( π/ − θ ′ ) ≤ θ ′ − θ ) /
2) sin(( θ ′ + θ ) / ≤ θ ′ − θ ) sin( θ ) ≤ θ ′ − θ ) sin( π/ − arcsin( 1 M )) ≤ θ ′ − θ ) cos(arcsin( 1 M )) ≤ θ ′ − θ ) r − M ≤ M sin( θ ′ − θ ) √ M − . This implies that (cid:13)(cid:13) ( µ + T + A ) − (cid:13)(cid:13) ≤ M | µ | sin( θ ′ − θ ) √ M − . This shows that the sector S belongs to ρ ( T + A ) and for any ε > (cid:13)(cid:13) ( µ + T + A ) − (cid:13)(cid:13) ≤ M ε | µ | for | arg ( µ ) | ≤ π/ − arcsin( 1 M ) + ε, with M ε = M sin( ε ) √ M − θ ′ − θ = ε . Clearly, M ε is independent of µ . Hence, T + A is m- ω -accretive, with ω = π/ − arcsin( b − b ). (cid:3) Remark . (1) As seen in the last paragraph of the proof, the condition (1.2) implies(1.3) at least for 0 ≤ b ≤
1. Thus [13, Theorem 4.1] is covered by Lemma 1.1.(2) If the assumptions of Lemma 1.1 are satisfied, we can see that
Re < tx + T x, Ax > ≥ x ∈ D ( T ). Therefore A ( t + T ) − is bounded accretive operator for any t > PERTURBATION RESULT OF M-ACCRETIVE LINEAR OPERATORS 5
Corollary 2.2.
Let T and A as in Lemma 1.1 obeying (1.3) . Then (1) − ( T + A ) generates contractive one-parameter semigroup for ≤ b ≤ . (2) − ( T + A ) generates contractive holomorphic one-parameter semigroup with angle ω = arcsin( b − b ) for b > . An application
One of interest is the operator-norm error bound estimate for the exponential Trotter-Kato product formula in the case of accretive perturbations, see [1, 10, 11] and [12] for ashort survey. Let A be a semibounded from below densely defined self-adjoint operatorand B an m-accretive operator in a Hilbert space H .In [1, Theorem 3.4] it has been shown that if B is A -bounded with lower bound < D (( A + B ) α ) ⊂ D ( A α ) ∩ D (( B ∗ ) α ) = { } for some α ∈ (0 . , (3.1)then there is a constant L α > (cid:13)(cid:13)(cid:13)(cid:0) e − tB/n e − tA/n (cid:1) n − e − t ( A + B ) (cid:13)(cid:13)(cid:13) ≤ L α ln nn α (3.2)and (cid:13)(cid:13)(cid:13)(cid:0) e − tA ∗ /n e − tB ∗ /n (cid:1) n − e − t ( A + B ) ∗ (cid:13)(cid:13)(cid:13) ≤ L α ln nn α (3.3)hold for some α ∈ (0 .
1] and n = 1 , , . . . uniformly in t ≥
0. Here T α denotes the fractionalpowers of an m-accretive operator, see [8, 9].The aim of the present result is to extend [1, Theorem 3.4]. This extension is accom-plished by replacing the relative boundedness by the assumption (1.3). More precisely,we have Theorem 3.1.
Let A be a semibounded from below densely defined self-adjoint operatorand B an m-accretive operator with (1.3) for some b > . Assume that (3.1) holds. Thenthere is a constant L α > such that the estimates (3.2) and (3.3) hold for some α ∈ (0 . and n = 1 , , . . . uniformly in t ≥ .Proof. From (1.3), we have for b > k Bx k ≤ a k Ax k , for all x ∈ D ( A ) , (3.4)with a = b <
1. Hence B is A -bounded with lower bound a <
1. Also, by lemma1.1, A + B is m- ω -accretive, with ω = π/ − arcsin( b − b ). Now, all assumptions of [1,Theorem 3.4] are fulfilled. Hence we obtain the desired result. (cid:3) Remark . It well known that, for an m-accretive operator T , the fractional powers T α are m-( απ ) / α ∈ (0 , / D ( T α ) = D ( T ∗ α ), see [9, Theorem 1.1].Since A , B and A + B are m-accretive operators, we deduce that D (( A + B ) ∗ α ) = D (( A + B ) α ) ⊂ D ( A α ) ∩ D ( B α ) = D ( A α ) ∩ D (( B ∗ ) α ) , for some α ∈ (0 , / α ∈ (0 , /
2[ (cf. [1, Theorem 4.1]).
M. BENHARRAT
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