aa r X i v : . [ m a t h . F A ] N ov Quasi-Sectorial Contractions
Valentin A. Zagrebnov
Universit´e de la M´editerran´ee (Aix-Marseille II) and Centre de PhysiqueTh´eorique - UMR 6207, Luminy-Case 907, 13288 Marseille Cedex 9, France
Abstract
We revise the notion of the quasi-sectorial contractions. Our main theorem estab-lishes a relation between semigroups of quasi-sectorial contractions and a class of m − sectorial generators. We discuss a relevance of this kind of contractions to thetheory of operator-norm approximations of strongly continuous semigroups. Key words:
Operator numerical range; m -sectorial generators; contractionsemigroups; quasi-sectorial contractions; holomorphic semigroups; semigroupoperator-norm approximations. PACS:
Let H be a separable Hilbert space and let T be a densely defined linearoperator with domain dom( T ) ⊂ H . Definition 1.1
The set of complex numbers: N ( T ) := { ( u, T u ) ∈ C : u ∈ dom( T ) , k u k = 1 } , is called the numerical range of the operator T . Remark 1.1 ( a ) It is known that the set N ( T ) is convex ( the Toeplitz-Hausdorfftheorem ) , and in general is neither open nor closed, even for a closed operator T . ( b ) Let ∆ := C \ N ( T ) be complement of the numerical range closure in thecomplex plane. Then ∆ is a connected open set except the special case, when N ( T ) is a strip bounded by two parallel straight lines. Email address: [email protected] (Valentin A. Zagrebnov).
Preprint submitted to Elsevier elow we use some important properties of this set, see e.g. [7, Ch.V], or [11,Ch.1.6]. Recall that dim(ran( T )) ⊥ =: def( T ) is called a deficiency (or defect )of a closed operator T in H . Proposition 1.1 ( i ) Let T be a closed operator in H . Then for any complexnumber z / ∈ N ( T ) , the operator ( T − zI ) is injective. Moreover, it has a closedrange ran( T − zI ) and a constant deficiency def( T − zI ) in each of connectedcomponent of C \ N ( T ) . ( ii ) If def( T − zI ) = 0 for z / ∈ N ( T ) , then ∆ is a subset of the resolvent set ρ ( T ) of the operator T and k ( T − zI ) − k ≤ z, N ( T )) . (1.1)( iii ) If dom( T ) is dense and N ( T ) = C , then T is closable, hence the adjointoperator T ∗ is also densely defined. Corollary 1.1
For a bounded operator T ∈ L ( H ) the spectrum σ ( T ) is asubset of N ( T ) . For unbounded operator T the relation between spectrum and numerical rangeis more complicated. For example, it may very well happen that σ ( T ) is notcontained in N ( T ), but for a closed operator T the essential spectrum σ ess ( T )is always a subset of N ( T ). The condition def( T − zI ) = 0 , z / ∈ N ( T ) inProposition 1.1 (ii) serves to ensure that for those unbounded operators onegets σ ( T ) ⊂ N ( T ) , (1.2)i.e., the same conclusion as in Corollary 1.1 for bounded operators. Definition 1.2
Operator T is called sectorial with semi-angle α ∈ (0 , π/ z = 0 if N ( T ) ⊆ S α := { z ∈ C : | arg z | ≤ α } . If, in addition, T is closed and there is z ∈ C \ S α such that it belongs to theresolvent set ρ ( T ), then operator T is called m -sectorial. Remark 1.2
Let T be m -sectorial with the semi-angle α ∈ (0 , π/ and thevertex at z = 0 . Then it is obvious that the operators aT and T b := T + b belong to the same sector S α for any non-negative parameters a, b ≥ . In fact N ( T b ) ⊆ S α + b , i.e. the operator T b has the vertex at z = b . Some of important properties of the m -sectorial operators are summarized bythe following 2 roposition 1.2 If T is m -sectorial in H , then the semigroup { U ( ζ ) := e − ζ T } ζ generated by the operator T :( i ) is holomorphic in the open sector { ζ ∈ S π/ − α } ;( ii ) is a contraction, i.e. N ( U ( ζ )) is a subset of the unit disc D r =1 := { z ∈ C : | z | ≤ } for { ζ ∈ S π/ − α } . The notion of the quasi-sectorial contractions was introduced in [4] to studythe operator-norm approximations of semigroups. In paper [3] this class ofcontractions appeared in analysis of the operator-norm error bound estimateof the exponential Trotter product formula for the case of accretive perturba-tions. Further applications of these contractions which, in particular, improvethe rate of convergence estimate of [4] for the Euler formula, one can find in[9], [2] and [1].
Definition 2.1
For α ∈ [0 , π/
2) we define in the complex plane C a closeddomain : D α := { z ∈ C : | z | ≤ sin α } ∪ { z ∈ C : | arg(1 − z ) | ≤ α and | z − | ≤ cos α } . This is a convex subset of the unit disc D r =1 , with ”angle” (in contrast to tangent ) touching of its boundary ∂ D r =1 at only one point z = 1, see Figure1. It is evident that D α ⊂ D β>α . Definition 2.2 ( Quasi-Sectorial Contractions [4]) A contraction C on theHilbert space H is called quasi-sectorial with semi-angle α ∈ [0 , π/
2) withrespect to the vertex at z = 1, if N ( C ) ⊆ D α .Notice that if operator C is a quasi-sectorial contraction, then I − C is an m - sectorial operator with vertex z = 0 and semi-angle α . The limits α = 0and α = π/ self-adjoint ) andto general contraction.The resolvent of an m -sectorial operator A , with semi-angle α ∈ (0 , π/
4] andvertex at z = 0, gives the first non-trivial (and for us a key ) example of aquasi-sectorial contraction. Proposition 2.1
Let A be m -sectorial operator with semi-angle α ∈ [0 , π/ and vertex at z = 0 . Then { F ( t ) := ( I + tA ) − } t ≥ is a family of quasi-sectorialcontractions which numerical ranges N ( F ( t )) ⊆ D α for all t ≥ . roof : First, by virtue of Proposition 1.1 (ii) we obtain the estimate: k F ( t ) k ≤ t dist(1 /t , − S α ) = 1 , (2.1)which implies that operators { F ( t ) } t ≥ are contractions with numerical ranges N ( F ( t )) ⊆ D r =1 .Next, by Remark 1.2 for all u ∈ H one gets ( u, F ( t ) u ) = ( v t , v t ) + t ( Av t , v t ) ∈ S α , where v t := F ( t ) u , i.e. for any t ≥ N ( F ( t )) ⊆ S α .Similarly, one finds that ( u, ( I − F ( t )) u ) = t ( v, Av ) + t ( Av, Av ) ∈ S α , i.e., N ( I − F ( t )) ⊆ S α . Therefore, for all t ≥ N ( F ( t )) ⊆ ( S α ∩ (1 − S α )) ⊂ D r =1 . (2.2)Moreover, since α ≤ π/
4, by Definition 2.1 we get ( S α ∩ (1 − S α )) ⊂ D α , i.e.for these values of α the operators { F ( t ) } t ≥ are quasi-sectorial contractionswith numerical ranges in D α . (cid:3) Now we are in position to prove the main
Theorem establishing a relation be-tween quasi-sectorial contraction semigroups and a certain class of m -sectorialgenerators. Theorem 2.1
Let A be an m -sectorial operator with semi-angle α ∈ [0 , π/ and with vertex at z = 0 . Then { e − t A } t ≥ is a quasi-sectorial contractionsemigroup with numerical ranges N ( e − t A ) ⊆ D α for all t ≥ . The proof of the theorem is based on a series of lemmata and on the numericalrange mapping theorem by Kato [8] (see also an important comment aboutthis theorem in [10]).
Proposition 2.2 [8]
Let f ( z ) be a rational function on the complex plane C ,with f ( ∞ ) = ∞ . Let for some compact and convex set E ′ ⊂ C the inversefunction f − : E ′ E ⊇ K , where K is a convex kernel of E , i.e., a subsetof E such that E is star-shaped relative to any z ∈ K .If C is an operator with numerical range N ( C ) ⊆ K , then N ( f ( C )) ⊆ E ′ . Notice that for a convex set E the corresponding convex kernel K = E . Lemma 2.1
Let f n ( z ) = z n be complex functions, for z ∈ C and n ∈ N .Then the sets f n ( D α ) are convex and domains f n ( D α ) ⊆ D α for any n ∈ N ,if α ≤ π/ . emma 2.2 (Euler formula) Let A be an m -sectorial operator. Then for t ≥ one gets the strong limit s − lim n →∞ ( F ( t/n )) n = e − tA . (2.3)The next section is reserved for the proofs. They refine and modify somelines of reasonings of the paper [4]. This concerns, in particular, a correctedproofs of Proposition 2.1 and Theorem 2.1 (cf. Theorem 2.1 of [4]), as well asreformulations and proofs of Propositions 2.2 and Lemma 2.1. (Lemma 2.1):Let { z : | z | ≤ sin α } ⊂ D α , then one gets | z n | ≤ sin α . Therefore, for themappings f n : z z n one obtains f n ( z ) ∈ D α for any n ≥ f n ( G α ) , n ≥ G α := { z : | arg( z ) | < ( π/ − α ) } ∩ { z : | arg( z + 1) | > ( π − α ) } ⊂ D α , (3.1)see Definition 2.1 and Figure 1.For 0 ≤ t ≤ cos α , two segments of tangent straight intervals: { ζ ± ( t ) = 1 + t e i ( π ∓ α ) } ≤ t ≤ cos α ⊂ ∂D α , are correspondingly upper ζ + ( t ) and lower ζ − ( t ) = ζ + ( t ) non-arc parts ofthe total boundary ∂D α ; they also coincide with a part of the boundary ∂ G α connected to the vertex z = 1.Now we proceed by induction. Let n = 1. Then one obviously obtain : f n =1 ( D α ) = D α . For n = 2 the boundary ∂f ( G α ) of domain f ( G α ) is aunion Γ ( α ) ∪ Γ ( α ) of the contourΓ ( α ) := { f ( ζ + ( t )) } ≤ t ≤ cos α ∪ { z : | z | ≤ sin α, arg( z ) = ( π − α ) } and its conjugate Γ ( α ). Since arg( ∂ t f ( ζ + ( t )) ≤ ( π − α ) for all 0 ≤ t ≤ cos α ,the contour { f ( ζ + ( t )) } ≤ t ≤ cos α ⊆ { z : | arg( z + 1) | > ( π − α ) } , see (3.1). The same is obviously true for the image of the lower branch ζ − ( t ).If α ≤ π/
4, one gets: 5up ≤ t ≤ cos α Im( f ( ζ + ( t ))) = Im( f ( ζ + ( t ∗ = (2 cos α ) − ))) (3.2)= 12 tan α < sin α cos α , where t ∗ = (2 cos α ) − ≤ cos α , and0 ≥ Re( f ( ζ + ( t ))) ≥ − sin α cos 2 α ≥ − sin α . Therefore, { f ( ζ + ( t )) } ≤ t ≤ cos α ⊆ D α . Since the same is also true for the im-age of the lower branch ζ − ( t ), we obtain f ( G α ) ⊂ D α and by consequence f n =2 ( D α ) = { w = z · z : z ∈ D α , z ∈ f n =1 ( D α ) } ⊂ D α , for α ≤ π/ n > f n ( D α ) ⊂ D α . Then the image of the ( n +1) − order mapping of domain D α is: f n +1 ( D α ) = { w = z · z n : z ∈ D α , z n ∈ f n ( D α ) } , and since f n ( D α ) ⊂ D α , we obtain f n +1 ( D α ) ⊂ D α by the same reasoning asfor n = 2. (cid:3) Remark 3.1
Let φ ( t ) := arg( ζ + ( t )) . Then cot( α + φ ( t )) = (cos α − t ) / sin α and sup ≤ t ≤ cos α Im( f n ( ζ + ( t ))) ≤ (1 − t ∗ n cos α + ( t ∗ n ) ) n/ (3.3) for sin( nφ ( t ∗ n )) = 1 . In the limit n → ∞ this implies that φ ( t ∗ n ) = π/ n + o ( n − ) , t ∗ n = π/ (2 n sin α ) + o ( n − ) and lim n →∞ sup ≤ t ≤ cos α Im( f n ( ζ + ( t ))) ≤ exp( − π cot α ) <
12 tan α. (3.4) By the same reasoning one gets the estimates similar to (3.3) and (3.4) for ζ − ( t )) . Hence, | Im( f n ( ζ ± ( t ))) | < Im( f n =1 ( ζ + ( t ))) < sin α cos α , cf. (3.2).Notice that in spite of the arc-part of the contour ∂D α shrinks in the limit n → ∞ to zero, we obtain lim n →∞ sup ≤ t ≤ cos α Re( f n ( ζ + ( t ))) = − exp( − π cot α ) , (3.5) for the left extreme point of the projection on the real axe ( sin( nφ ( t ∗ n )) = 1 ) ofthe image f n ( D α ) . Since exp( − π cot α ) < sin α , for α ≤ π/ , the arguments(3.4) and (3.5) bolster the conclusion of the Lemma 2.1. roof (Lemma 2.2):By (2.1) we have for λ > k ( λI + A ) − k < λ − , (3.6)and since A is m -sectorial, we also get that ( −∞ , ⊂ ρ ( A ). Then the Hille-Yosida theory ensures the existence of the contraction semigroup { e − t A } t ≥ ,and the standards arguments (see e.g. [7, Ch.V], or [11, Ch.1.1]) yield theconvergence of the Euler formula (2.3) in the strong topology. (cid:3) Proof (Theorem 2.1):Take f ( z ) = z and the compact convex set E ′ := f ( D α ) ⊆ D α , see Lemma2.1. Since the set E := f − ( E ′ ) = D α ∪ ( − D α ) is convex , its convex kernel K exists and K = E . Then by Proposition 2.2 we obtain that N ( f ( C )) ⊆ E ′ ⊆ D α , if the numerical range N ( C ) ⊆ K .Let contraction C := ( I + t A/ − = F ( t/ t ≥ N ( C ) ⊆ D α and since D α ⊂ E , we can choose K = E .Then by the Kato numerical range mapping theorem (Proposition 2.2) we get: N ( f ( C ) = F ( t/ ) ⊆ E ′ ⊆ D α . (3.7)Similarly, take the contraction C := F ( t/ . Since (3.7) is valid for any t ≥ t t/
2. Then by definition of K one has N ( F ( t/ ) ⊆ D α ⊆ K .Now again the Proposition 2.2 implies: N ( f ( C ) = F ( t/ ) ⊆ E ′ ⊆ D α . (3.8)Therefore, we obtain N ( F b ( t/ n ) n ) ⊆ D α , for any n ∈ N . By Lemma 2.2 thisyields lim n →∞ ( u, ( I + t A/ n ) − n u ) = ( u, e − t A u ) ∈ D α , for any unit vector u ∈ H . Therefore, the numerical ranges of the contractionsemigroup N ( e − t A ) ⊆ D α for all t ≥
0, if it is generated by m -sectorial operatorwith the semi-angle α ∈ [0 , π/
4] and with the vertex at z = 0. (cid:3)
1. Notice that Definition 2.2 of quasi-sectorial contractions C is quite restric-tive comparing to the notion of general contractions, which demands only N ( C ) ⊆ D . For the latter case one has a well-known Chernoff lemma [5]: k ( C n − e n ( C − I ) ) u k ≤ n / k ( C − I ) u k , u ∈ H , n ∈ N , (4.1)7hich is not even a convergent bound. For quasi-sectorial contractions we canobtain a much stronger estimate [4]: (cid:13)(cid:13)(cid:13) C n − e n ( C − I ) (cid:13)(cid:13)(cid:13) ≤ M n − / , n ∈ N , (4.2)convergent to zero in the uniform topology when n → ∞ . Notice that the rateof convergence n − / obtained in [4] with help of the Poisson representation and the
Tchebychev inequality is not optimal. In [9], [2] and [1] this estimatewas improved up to the optimal rate O ( n − ), which one can easily verify for aparticular case of self-adjoint contractions (i.e. α = 0) with help of the spectralrepresentation.The inequality (4.2) and its further improvements are based on the followingimportant result about the upper bound estimate for the case of quasi-sectorial contractions: Proposition 4.1 If C is a quasi-sectorial contraction on a Hilbert space H with semi-angle ≤ α < π/ , i.e. the numerical range N ( C ) is a subset of thedomain D α , then k C n ( I − C ) k ≤ Kn + 1 , n ∈ N . (4.3)For the proof see Lemma 3.1 of [4].2. Another application of quasi-sectorial contractions generalizes the Chernoffsemigroup approximation theory [5], [6] to the operator-norm approximations[4]. Proposition 4.2
Let { Φ( s ) } s ≥ be a family of uniformly quasi-sectorial con-tractions on a Hilbert space H , i.e. such that there exists < α < π/ and N (Φ( s )) ⊆ D α , for all s ≥ . Let X ( s ) := ( I − Φ( s )) /s , and let X be a closed operator with non-empty resolvent set, defined in aclosed subspace H ⊆ H . Then the family { X ( s ) } s> converges, when s → +0 ,in the uniform resolvent sense to the operator X if and only if lim n →∞ (cid:13)(cid:13)(cid:13) Φ( t/n ) n − e − tX P (cid:13)(cid:13)(cid:13) = 0 , for t > . (4.4) Here P denotes the orthogonal projection onto the subspace H .
3. We conclude by application of Theorem 2.1 and Proposition 4.1 to the Eulerformula [4], [9], [2]. 8 roposition 4.3 If A is an m -sectorial operator in a Hilbert space H , withsemi-angle α ∈ [0 , π/ and with vertex at z = 0 , then lim n →∞ (cid:13)(cid:13)(cid:13) ( I + tA/n ) − n − e − tA (cid:13)(cid:13)(cid:13) = 0 , t ∈ S π/ − α . Moreover, uniformly in t ≥ t > one has the error estimate: (cid:13)(cid:13)(cid:13) ( I + tA/n ) − n − e − tA (cid:13)(cid:13)(cid:13) ≤ O (cid:16) n − (cid:17) , n ∈ N . Acknowledgements
I would like to thank Professor Mitsuru Uchiyama for a useful remark indi-cating a flaw in our arguments in Section 2 of [4] , revision of this part of thepaper [4] is done in the present manuscript. I also thankful to Vincent Cachiaand Hagen Neidhardt for a pleasant collaboration.
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Leuven Notes inMathematical and Theoretical Physics. Series A: Mathematical Physics, 10.Leuven University Press, Leuven, 2003. -sin ΓΓ a β β r α α D B a ** a * i-i0 A Σ r -sin Fig. 1. Illustration of the set D α (= Σ a ∗ shaded domain) with boundary ∂D α = Γ a ∗ ,where a ∗ = sin α , as well as of our choice of the contour Γ r in the resolvent set ρ ( C ),where r = sin β > a ∗ . The contour Γ r consists of two segments of tangent straightlines (1 , A ) and (1 , B ) and the arc ( A, B ) of radius r . The dotted circle ∂ D r =1 / corresponds to the set of tangent points for different values of α ∈ [0 , π/2].