aa r X i v : . [ m a t h . F A ] J a n A NOTE ON APPROXIMATION OF OPERATOR SEMIGROUPS
JOCHEN GL ¨UCK
Abstract.
Let A be a bounded linear operator and P a bounded linearprojection on a Banach space X . We show that the operator semigroup( e t ( A − kP ) ) t ≥ converges to a semigroup on a subspace of X as k → ∞ andwe compute the limit semigroup. Introduction and discussion of the main result
To motivate the content of this note, let (Ω , Σ , µ ) be a σ -finite measure space and( e tA ) t ≥ a positive C -semigroup on X := L p (Ω , Σ , µ ) for some p ∈ [1 , ∞ ). If B ⊂ Ωis a measurable set, if P is the projection on X which is given by multiplicationwith the indicator function B and if Q := 1 − P , then for all t > k →∞ e t ( A − kP ) and lim k →∞ (cid:0) e tk A Q (cid:1) k (1.1)exist with respect to the strong operator topology and coincide [1, Lemma 4.1and Theorem 5.3]. The limit can be shown to be a certain degenerate operatorsemigroup [1, p. 431–432] which we might refer to as a sort of absorption semigroup .Usually, the latter term is used to describe the semigroup governed by an abstractCauchy problem which models a diffusion process with a potential (= absorptionterm). For the connection of such absorption semigroups to the limit semigroupin (1.1) we refer the reader to [1]; see also [8, 9] for some additional backgroundinformation.When considering the above convergence result, a number of potential general-isations immediately comes into ones mind; for example, one could try to replacethe projection P , which is given by multiplication with an indicator function, by amore general projection. One could also try to consider more general spaces and/orto omit the positivity assumption. We refer to [2, 7, 5, 6] and the references thereinfor a number of positive and negative results in this direction. It is also worth-while pointing out that the second limit in (1.1) is closely related to the so-called Quantum Zeno Effect ; we refer to [3, 4] and the references therein for more details.In this note we are concerned with the existence of the limits in (1.1) in thecase where A is a bounded linear operator (without any additional properties) onan arbitrary Banach space and where P is an arbitrary bounded linear projection.For this case, the second limit in (1.1) was shown by Matolcsi and Shvidkoy toalways exist with respect to the strong operator topology; moreover they identifiedthe limit semigroup and thus proved the following theorem [7, Theorem 1]: Theorem 1.1 (Matolcsi, Shvidkoy) . Let A be a bounded linear operator and Q abounded linear projection on a Banach space X . For each x ∈ X and each t ≥ we have (cid:0) e tk Q (cid:1) k x → e tQAQ Qx as k → ∞ , where the convergence is uniform with respect to t on bounded subsets of [0 , ∞ ) . Date : January 26, 2015.
Key words and phrases. absorption semigroup; degenerate semigroup; projection; boundedgenerator; resolvent estimates; contour integration.
As the generator A is bounded it is natural to ask whether the limit exists even inthe operator norm topology. In Section 3 at the end of this note we will prove thatthis is indeed true. Our focus, however, is on another question, namely whetherthe limit lim k →∞ e t ( A − kP ) always exists if A is bounded. Our main result gives anaffirmative answer to this question: Theorem 1.2.
Let A be a bounded linear operator and P a bounded linear projec-tion on a complex Banach space X . Define Q := 1 − P and let z ∈ C . For all t > we have e t ( A + zP ) → e tQAQ Q as Re z → −∞ with respect to the operator norm, and the convergence is uniform with respect to t on compact subsets of (0 , ∞ ) . The proof of Theorem 1.2, which we present in Section 2, uses only elementarymethods, but it requires a careful analysis of the spectral properties of A + zP .In fact we will obtain a bit more information about the convergence in the abovetheorem, including an explicit estimate (for undefined notation we refer to the endof the introduction): Remark 1.3.
For t in any fixed compact subset of (0 , ∞ ), the convergence inTheorem 1.2 has at least a linear rate.More precisely, the following holds: Let 0 < T < T and define R := 2( k A k + δ ) k P − Q k , where δ > R is strictlylarger than the spectral radius of QAQ . Then for every z ∈ C with Re z < − R and all t ∈ [ T , T ] we have k e t ( A + zP ) − e tQAQ Q k ≤ C e T Re z + C | z | − R , where C := Re T R k A k + δδ k P k sup | λ | = R k I + AQ R ( λ, QAQ ) k and C := Re T R δ ;here, R ( λ, QAQ ) denotes the resolvent of QAQ at λ . Note that C can be furtherestimated by means of the Neumann series if we chose δ > R > k QAQ k .A few further remarks are in order. Remarks 1.4. (a) Using complexifications one immediately obtains an analoguesresult for real Banach spaces.(b) The operator family ( e tQAQ Q ) t ≥ is a (norm continuous) C -semigroup on QX = ker P . Hence, the semigroup ( e t ( A + zP ) ) t ≥ converges to 0 on the range of P and to a semigroup with generator QAQ on the kernel of P .(c) Using the power series expansion of the exponential function one obtainsseveral ways to write down the limit semigroup; in fact, one has e tQAQ Q = e tQA Q = Qe tAQ = Qe tQAQ = Qe tQAQ Q for every t ∈ (0 , ∞ ).(d) The assertion of Theorem 1.2 is obviously false for t ≤
0; just consider X = C , A = 0 and P = 1 to see this.(e) A glance at the proof of Theorem 1.2 in the next section reveals that onecan show similar results for analytic functions f other than exp, provided that f satisfies certain decay properties. Since our focus is on operator semigroups, weshall not discuss this in detail.It seems to be unclear what happens to the assertion of Theorem 1.2 if we con-sider C -semigroups with unbounded generator A . Of course one expects that someadditional conditions are necessary in this case, since otherwise it might happen NOTE ON APPROXIMATION OF OPERATOR SEMIGROUPS 3 that
QAQ has only a very small domain which need not be dense in the range of Q ;moreover, QAQ might not even be closed. Concerning the related Theorem 1.1, wealso point out that the strong limit lim k →∞ (cid:0) e tk A Q (cid:1) k never exists for all boundedlinear projections Q on X unless A is bounded [6, Theorem 2.1]. In any case, theproof of Theorem 1.2 which we present below relies heavily on the boundednessof A . We therefore leave it as an open problem to analyse the case of unboundedgenerators.Before we give a proof of our main result in the next section, let us briefly fixsome notation. If X is a complex Banach space, then we denote by L ( X ) the spaceof all bounded linear operators on X . The spectrum of an operator A ∈ L ( X ) isdenoted by σ ( A ) and for every λ ∈ C \ σ ( A ), R ( λ, A ) := ( λ − A ) − denotes theresolvent of A at λ . For r ≥ z ∈ C we denote by B r ( z ) := { µ ∈ C | | µ − z | < r } the open disk in C with center z and radius r and by B r ( z ) := { µ ∈ C | | µ − z | ≤ r } the closed disk in C with center z and radius r .2. Proof of the main result
In this section we prove our main result. Throughout the section we may assumethat X = { } and we let 0 < T < T . We want to find a norm estimate for thedifference e t ( A + zP ) − e tQAQ Q , where t ∈ [ T , T ] and where Re z → −∞ . To thisend we employ the functional calculus for analytic functions and write e t ( A + zP ) − e tQAQ Q = 12 πi I γ e tλ [ R ( λ, A + zP ) − R ( λ, QAQ ) Q ] dλ (2.1)for an appropriate path γ which encircles both the spectra of A + zP and QAQ .Of course we could choose γ to be a sufficiently large circle, but this is too crudeto obtain a reasonable estimate. Hence, our first task is to localize the spectrum of A + zP more precisely in order to find a good choice for γ .To do this, we first show a simple auxiliary result about the resolvent of ourprojection P . Lemma 2.1.
Let z ∈ C . (a) We have σ ( zP ) ⊂ { , z } and for every λ ∈ C \ { , z } , the resolvent of zP is given by R ( λ, zP ) = λ − zQλ ( λ − z ) (2.2)= 12 (cid:0) λ + 1 λ − z + z ( P − Q ) λ ( λ − z ) (cid:1) . (2.3)(b) Consider a non-negative parameter α ≥ and the radius r α := 2 α k P − Q k .If λ ∈ C is contained in none of the closed disks B r α (0) and B r α ( z ) , thenthe resolvent R ( λ, zP ) fulfils the estimate α kR ( λ, zP ) k < . Loosely speaking, the estimate in (b) says that the resolvent R ( λ, zP ) decreaseslinearly as λ tends away from the points 0 and z . Proof of Lemma 2.1. (a) We clearly have σ ( zP ) ⊂ { , z } since σ ( P ) ⊂ { , } . Therepresentation formulas for the resolvent can be verified by a simple computation.(b) Assume that λ B r α (0) ∪ B r α ( z ); then | λ | > r α ≥ α and | λ − z | > r α ≥ α .Moreover, at least one of the numbers | λ | and | λ − z | is no less than | z | since wehave | z | ≤ | λ | + | λ − z | . Since the other one is no less than r α = 2 α k P − Q k , we JOCHEN GL¨UCK conclude that | λ || λ − z | ≥ α | z |k P − Q k . Hence we conclude from the resolventrepresentation formula (2.3) in (a) that α kR ( λ, zP ) k ≤ (cid:0) α | λ | + α | λ − z | + α | z |k P − Q k| λ || λ − z | (cid:1) < , which proves the assertion. (cid:3) Now we can achieve our first goal and obtain very precise information on theposition of the spectrum σ ( A + zP ); we also obtain a Neumann type series repre-sentation for the resolvent: Proposition 2.2.
Let z ∈ C and consider the radius r := 2 k A kk P − Q k . (a) The spectrum of A + zP is contained in the union of the two closed disks B r (0) and B r ( z ) . (b) If λ ∈ C is contained in none of the discs B r (0) and B r ( z ) , then R ( λ, A + zP ) = ∞ X k =0 (cid:0) R ( λ, zP ) A (cid:1) k R ( λ, zP ) , where the series converges absolutely with respect to the operator norm.Proof. For every λ B r (0) ∪ B r ( z ) the equation λ − ( A + zP ) = ( λ − zP ) − A = ( λ − zP ) (cid:0) − R ( λ, zP ) A (cid:1) . holds. If we apply Lemma 2.1(b) with the parameter α := k A k , then we obtainthe estimate k A kkR ( λ, zP ) k <
1, and thus (a) and (b) follow by employing theNeumann series. (cid:3)
Part (a) of the above proposition suggests a strategy to estimate the expressionin (2.1): we rewrite the contour integral in (2.1) as a sum of contour integralsaround B r (0) and B r ( z ); the path of integration around the first disk should alsobe sufficiently large to encircle the spectrum of QAQ . Let us therefore introducethe following notation:
Notation 2.3.
For the rest of this section we use the following notation:(a) Let r := 2 k A kk P − Q k as in Proposition 2.2.(b) Let R := 2( k A k + δ ) k P − Q k > r , where δ > B R (0) contains the spectrum of QAQ .If | z | > R , then the circles with radius R around 0 and z do not intersect;using the information about the spectrum of A + zP that we obtained in Proposi-tion 2.2(a), we can therefore rewrite (2.1) as e t ( A + zP ) − e tQAQ Q = 12 πi I | λ − z | = R e tλ R ( λ, A + zP ) dλ + 12 πi I | λ | = R e tλ [ R ( λ, A + zP ) − R ( λ, QAQ ) Q ] dλ, (2.4)where the paths of integration are parametrized with positive orientation. Thespectra of A + zP and QAQ and the paths of integration in formula (2.4) areshown in Figure 2.1.Our goal is now to estimate both integrals in formula (2.4). Let us start withthe first integral, which turns out to be rather easy: since t ≥ T , the exponentialterm within the integral ensures a fast decay as Re z → −∞ provided that we cancontrol the resolvent. This is the content of the following proposition. Proposition 2.4.
Let z ∈ C , Re z < − R . (a) We have kR ( λ, A + zP ) k ≤ δ for all λ ∈ C with | λ − z | = R . NOTE ON APPROXIMATION OF OPERATOR SEMIGROUPS 5 σ ( A + zP ) σ ( QAQ ) R i R r RzrR Figure 2.1.
Spectra of A + zP and QAQ ; the dashed circles depictthe paths of integration in formula (2.4).(b)
For all t ∈ [ T , ∞ ) we have k πi I | λ − z | = R e tλ R ( λ, A + zP ) dλ k ≤ Re T (Re z + R ) δ . Proof. (a) To prove (a) we do not really need that Re z < − R but only thatthe disks B R (0) and B R ( z ) do not intersect. Let λ ∈ C with | λ − z | = R . InProposition 2.2(b) we proved the resolvent representation formula R ( λ, A + zP ) = ∞ X k =0 (cid:0) R ( λ, zP ) A (cid:1) k R ( λ, zP ) . From Lemma 2.1(b) we obtain (with α := k A k + δ and by approximating the circle ∂B r α ( z ) from the outside) the estimate ( k A k + δ ) kR ( λ, zP ) k ≤
1. Plugging thisinto the above representation formula for R ( λ, A + zP ) we compute kR ( λ, A + zP ) k ≤ ∞ X k =0 (cid:0) k A kk A k + δ (cid:1) k k A k + δ = 1 δ . (b) Assertion (b) readily follows from (a) since Re λ ≤ Re z + R < λ inthe path of integration. (cid:3) Part (b) of the above proposition shows that the first integral in (2.4) exhibitsthe decay rate claimed in Remark 1.3. It therefore remains to consider the secondintegral. Here, the exponential term is bounded below and above, so we have toshow that the difference R ( λ, A + zP ) −R ( λ, QAQ ) Q converges to 0 as Re z → −∞ .To do this, we represent the difference of both resolvents as a bounded multiple of R ( λ, A + zP ) P ; this latter term can then easily be seen to converge to 0. Proposition 2.5.
Let z ∈ C , | z | > R . (a) There is a bounded function M : ∂B R (0) → L ( X ) , not depending on z ,such that R ( λ, A + zP ) − R ( λ, QAQ ) Q = R ( λ, A + zP ) P M ( λ ) for all λ with | λ | = R . (b) We have kR ( λ, A + zP ) P k ≤ k A k + δδ k P k| λ − z | for all λ with | λ | = R .Proof. (a) Using that Q commutes with R ( λ, QAQ ), we can verify by a brief com-putation that R ( λ, A + zP ) − R ( λ, QAQ ) Q = R ( λ, A + zP ) P (cid:0) I + AQ R ( λ, QAQ ) (cid:1) JOCHEN GL¨UCK for all λ which are contained in the resolvent sets of both A + zP and QAQ . Hence,we simply have to define M ( λ ) := I + AQ R ( λ, QAQ ) for all λ ∈ ∂B R (0).(b) Let | λ | = R . If we choose α = k A k + δ in Lemma 2.1(b) (and approximatethe circle ∂B r α (0) from the outside) we obtain kR ( λ, zP ) k ≤ k A k + δ . Plugging thisinto the resolvent representation formula from Proposition 2.2(b) yields kR ( λ, A + zP ) P k ≤ k A k + δδ kR ( λ, zP ) P k . However, using formula (2.2) in Lemma 2.1(a), we can easily see that the operator R ( λ, zP ) P coincides with Pλ − z . This proves the asserted estimate. (cid:3) As a consequence we obtain the desired estimate for the second integral in for-mula (2.4):
Proposition 2.6.
There is a number
C > , independent of z , such that k πi I | λ | = R e tλ [ R ( λ, A + zP ) − R ( λ, QAQ ) Q ] dλ k ≤≤ C Re T R k A k + δδ k P k| z | − R . for all t ∈ [ − T , T ] and for all z ∈ C with | z | > R .Proof. The assertion follows immediately from Proposition 2.5 if we define C =sup | λ | = R k M ( λ ) k . Note that if M is chosen as in the proof of Proposition 2.5, then C has the value C = sup | λ | = R k I + AQ R ( λ, QAQ ) k . (cid:3) This completes the proof of our main theorem, and the choice of the constant C in the above proof also gives us the estimate claimed in Remark 1.3.It is interesting to note that the convergence of the first integral in (2.4) is due tothe decay of the exponential function as Re z → −∞ , while the convergence of thesecond integral only relies on the decay of the difference R ( λ, A + zP ) −R ( λ, QAQ ) Q as | z | → ∞ .3. Operator norm convergence in Matolcsi’s and Shvidkoy’s Theorem
In this final section we briefly demonstrate that in Theorem 1.1 the convergencehappens in fact with respect to the operator norm. At first glance, one might expectthat we have to reprove the theorem, possibly by another method or with betterestimates, to obtain this result. But in fact, things are much easier: we will usea simple lifting argument to derive operator norm convergence from the fact thatwe already know about the strong convergence. By the way, the theorem of courseholds for negative times t , too. Theorem 3.1.
Let A be a bounded linear operator and Q a bounded linear projec-tion on a Banach space X . For all t ∈ R we have (cid:0) e tk A Q (cid:1) k → e tQAQ Q as k → ∞ with respect to the operator norm; moreover, the convergence is uniform with respectto t in bounded subsets of R .Proof. First note that Theorem 1.1 obviously remains true for t ∈ R (consider thenegative generator − A to handle the case of negative times).Denote by B X the closed unit ball in X and let ˆ X := ℓ ∞ ( B X ; X ) be the space ofall bounded maps from B X to X . Clearly, ˆ X is a Banach space when endowed with NOTE ON APPROXIMATION OF OPERATOR SEMIGROUPS 7 the supremum norm. For every B ∈ L ( X ), define ˆ B ∈ L ( ˆ X ) by ˆ B ( y x ) := ( By x )for all ( y x ) = ( y x ) x ∈ B X ∈ ˆ X ; the mapping L ( X ) → L ( ˆ X ) , B ˆ B is an isometric unital Banach algebra homomorphism. In particular, ˆ Q is a projec-tion on ˆ X .Fix T > x := ( x ) x ∈ B X ∈ ˆ X be the identity map from B X to X . Forevery B ∈ L ( X ) we have k B k = k ˆ B ˆ x k . Hence,sup t ∈ [ − T,T ] k (cid:0) e tk A Q (cid:1) k − e tQAQ Q k = sup t ∈ [ − T,T ] k (cid:0) e tk ˆ A ˆ Q (cid:1) k ˆ x − e t ˆ Q ˆ A ˆ Q ˆ Q ˆ x k , and the latter term converges to 0 as k → ∞ according to Theorem 1.1 (respectively,its version for t ∈ R ). (cid:3) References [1] W. Arendt and C. J. K. Batty. Absorption semigroups and Dirichlet boundary conditions.
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E-mail address : [email protected]@uni-ulm.de