We revise the notion of the quasi-sectorial contractions. Our main theorem
establishes 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 the theory of operator-norm approximations of strongly
continuous semigroups

Hal Id Hal

Valentin A. Zagrebnov

Valentin Zagrebnov

English

arXiv

CiteSeerX

Quasi-Sectorial Contractions

AbstractWe revise the notion of the quasi-sectorial contractions. Our main theorem establishes 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 the theory of operator-norm approximations of strongly continuous semigroups

Zagrebnov, Valentin A.

Elsevier - Publisher Connector 

Quasi-sectorial contractions 

We revise the notion of the quasi-sectorial contractions. Our main theorem establishes 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 the theory of operator-norm approximations of strongly continuous semigroups

Zagrebnov, Valentin, A.

HAL AMU

Quasi-sectorial contractionsValentin ZagrebnovTo cite this version:Valentin Zagrebnov. Quasi-sectorial contractions. Journal of Functional Analysis, Elsevier,2008, 254 (9), pp.2503-2511. <10.1016/j.jfa.2007.10.010>. <hal-00184941>HAL Id: hal-00184941https://hal.archives-ouvertes.fr/hal-00184941Submitted on 4 Nov 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Quasi-Sectorial ContractionsValentin A. ZagrebnovUniversite´ de la Me´diterrane´e (Aix-Marseille II) and Centre de PhysiqueThe´orique - UMR 6207, Luminy-Case 907, 13288 Marseille Cedex 9, FranceAbstractWe revise the notion of the quasi-sectorial contractions. Our main theorem estab-lishes a relation between semigroups of quasi-sectorial contractions and a class ofm−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: 47A55, 47D03, 81Q101 Sectorial OperatorsLet 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, Tu) ∈ C : u ∈ dom(T ), ‖u‖ = 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 operatorT .(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, whenN(T ) is a strip bounded by two parallel straight lines.Email address: zagrebnov@cpt.univ-mrs.fr (Valentin A. Zagrebnov).Preprint submitted to Elsevier ScienceBelow 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 setρ(T ) of the operator T and‖(T − zI)−1‖ ≤1dist(z, N(T )). (1.1)(iii) If dom(T ) is dense and N(T ) 6= 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 σess(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 onegetsσ(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, pi/2)and a vertex at z = 0 ifN(T ) ⊆ Sα := {z ∈ C : | arg z| ≤ α} .If, in addition, T is closed and there is z ∈ C \ Sα 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, pi/2) and thevertex at z = 0. Then it is obvious that the operators aT and Tb := T + bbelong to the same sector Sα for any non-negative parameters a, b ≥ 0. In factN(Tb) ⊆ Sα + b, i.e. the operator Tb has the vertex at z = b.Some of important properties of the m-sectorial operators are summarized bythe following2Proposition 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 ∈ Spi/2−α};(ii) is a contraction, i.e. N(U(ζ)) is a subset of the unit disc Dr=1 := {z ∈C : |z| ≤ 1} for {ζ ∈ Spi/2−α}.2 Quasi-Sectorial Contractions and Main TheoremThe 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, pi/2) we define in the complex plane C a closeddomain:Dα := {z ∈ C : |z| ≤ sin α} ∪ {z ∈ C : | arg(1− z)| ≤ α and |z − 1| ≤ cos α}.This is a convex subset of the unit disc Dr=1, with ”angle” (in contrast totangent) touching of its boundary ∂Dr=1 at only one point z = 1, see Figure1. It is evident that Dα ⊂ Dβ>α.Definition 2.2 (Quasi-Sectorial Contractions [4]) A contraction C on theHilbert space H is called quasi-sectorial with semi-angle α ∈ [0, pi/2) withrespect to the vertex at z = 1, if N(C) ⊆ Dα.Notice that if operator C is a quasi-sectorial contraction, then I − C is anm-sectorial operator with vertex z = 0 and semi-angle α. The limits α = 0and α = pi/2 correspond, respectively, to non-negative (i.e. self-adjoint) andto general contraction.The resolvent of an m-sectorial operator A, with semi-angle α ∈ (0, pi/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, pi/4]and vertex at z = 0. Then {F (t) := (I+tA)−1} t≥0 is a family of quasi-sectorialcontractions which numerical ranges N(F (t)) ⊆ Dα for all t ≥ 0.3Proof : First, by virtue of Proposition 1.1 (ii) we obtain the estimate:‖F (t)‖ ≤1t dist(1/t , −Sα)= 1 , (2.1)which implies that operators {F (t)} t≥0 are contractions with numerical rangesN(F (t)) ⊆ Dr=1.Next, by Remark 1.2 for all u ∈ H one gets (u, F (t)u) = (vt, vt) + t(Avt, vt) ∈Sα, where vt := F (t)u, i.e. for any t ≥ 0 the numerical range N(F (t)) ⊆ Sα.Similarly, one finds that (u, (I − F (t))u) = t(v, Av) + t2(Av,Av) ∈ Sα, i.e.,N(I − F (t)) ⊆ Sα. Therefore, for all t ≥ 0 we obtain:N(F (t)) ⊆ (Sα ∩ (1− Sα)) ⊂ Dr=1 . (2.2)Moreover, since α ≤ pi/4, by Definition 2.1 we get (Sα ∩ (1− Sα)) ⊂ Dα, i.e.for these values of α the operators {F (t)} t≥0 are quasi-sectorial contractionswith numerical ranges in Dα. ¤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, pi/4]and with vertex at z = 0. Then {e−t A} t≥0 is a quasi-sectorial contractionsemigroup with numerical ranges N(e−t A) ⊆ Dα for all t ≥ 0.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−1 : E ′ 7→ 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 fn(z) = zn be complex functions, for z ∈ C and n ∈ N.Then the sets fn(Dα) are convex and domains fn(Dα) ⊆ Dα for any n ∈ N,if α ≤ pi/4.4Lemma 2.2 (Euler formula) Let A be an m-sectorial operator. Then for t ≥ 0one gets the strong limits− limn→∞(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 2.1.3 ProofsProof (Lemma 2.1):Let {z : |z| ≤ sin α} ⊂ Dα, then one gets |zn| ≤ sin α. Therefore, for themappings fn : z 7→ zn one obtains fn(z) ∈ Dα for any n ≥ 1.Thus, it rests to check the same property only for images fn(Gα), n ≥ 1 of thesub-domain:Gα :={z : | arg(z)| <(pi/2− α)} ∩ {z : | arg(z + 1)| > (pi − α)}⊂Dα, (3.1)see Definition 2.1 and Figure 1.For 0 ≤ t ≤ cos α, two segments of tangent straight intervals:{ζ±(t) = 1 + t ei(pi∓α)}0≤t≤cos α ⊂ ∂Dα,are correspondingly upper ζ+(t) and lower ζ−(t) = ζ+(t) non-arc parts ofthe total boundary ∂Dα; they also coincide with a part of the boundary ∂Gαconnected to the vertex z = 1.Now we proceed by induction. Let n = 1. Then one obviously obtain :fn=1(Dα) = Dα. For n = 2 the boundary ∂f2(Gα) of domain f2(Gα) is aunion Γ2(α) ∪ Γ2(α) of the contourΓ2(α) := {f2(ζ+(t))}0≤t≤cos α ∪ {z : |z| ≤ sin2 α, arg(z) = (pi − 2α)}and its conjugate Γ2(α). Since arg(∂tf2(ζ+(t)) ≤ (pi−α) for all 0 ≤ t ≤ cos α,the contour{f2(ζ+(t))}0≤t≤cos α ⊆ {z : | arg(z + 1)| > (pi − α)},see (3.1). The same is obviously true for the image of the lower branch ζ−(t).If α ≤ pi/4, one gets:5sup0≤t≤cos αIm(f2(ζ+(t))) = Im(f2(ζ+(t∗ = (2 cos α)−1))) (3.2)=12tan α < sin α cos α ,where t∗ = (2 cos α)−1 ≤ cos α, and0 ≥ Re(f2(ζ+(t))) ≥ − sin2 α cos 2α ≥ − sin α .Therefore, {f2(ζ+(t))}0≤t≤cos α ⊆ Dα. Since the same is also true for the im-age of the lower branch ζ−(t), we obtain f2(Gα) ⊂ Dα and by consequencefn=2(Dα) = {w = z · z : z ∈ Dα, z ∈ fn=1(Dα)} ⊂ Dα, for α ≤ pi/4.Now let n > 2 and suppose that fn(Dα) ⊂ Dα. Then the image of the (n+1) −order mapping of domain Dα is:fn+1(Dα) = {w = z · zn : z ∈ Dα, zn ∈ fn(Dα)},and since fn(Dα) ⊂ Dα, we obtain fn+1(Dα) ⊂ Dα by the same reasoning asfor n = 2. ¤Remark 3.1 Let φ(t) := arg(ζ+(t)). Then cot(α + φ(t)) = (cos α − t)/ sin αandsup0≤t≤cos αIm(fn(ζ+(t))) ≤ (1− 2t∗n cos α + (t∗n)2)n/2 (3.3)for sin(nφ(t∗n)) = 1. In the limit n → ∞ this implies that φ(t∗n) = pi/2n +o(n−1), t∗n = pi/(2n sin α) + o(n−1) andlimn→∞sup0≤t≤cos αIm(fn(ζ+(t))) ≤ exp(−12pi cot α) <12tan α. (3.4)By the same reasoning one gets the estimates similar to (3.3) and (3.4) forζ−(t)). Hence, |Im(fn(ζ±(t)))| < Im(fn=1(ζ+(t))) < sin α cos α, cf. (3.2).Notice that in spite of the arc-part of the contour ∂Dα shrinks in the limitn →∞ to zero, we obtainlimn→∞sup0≤t≤cos αRe(fn(ζ+(t))) = − exp(−pi cot α), (3.5)for the left extreme point of the projection on the real axe (sin(nφ(t∗n)) = 1) ofthe image fn(Dα). Since exp(−pi cot α) < sin α, for α ≤ pi/4, the arguments(3.4) and (3.5) bolster the conclusion of the Lemma 2.1.6Proof (Lemma 2.2):By (2.1) we have for λ > 0‖(λI + A)−1‖ < λ−1 , (3.6)and since A is m-sectorial, we also get that (−∞, 0) ⊂ ρ(A). Then the Hille-Yosida theory ensures the existence of the contraction semigroup {e−t A} t≥0,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. ¤Proof (Theorem 2.1):Take f(z) = z2 and the compact convex set E ′ := f(Dα) ⊆ Dα, see Lemma2.1. Since the set E := f−1(E ′) = Dα ∪ (−Dα) is convex, its convex kernel Kexists and K = E. Then by Proposition 2.2 we obtain that N(f(C)) ⊆ E ′ ⊆Dα, if the numerical range N(C) ⊆ K.Let contraction C1 := (I + t A/2)−1 = F (t/2). Since by Proposition 2.1 forany t ≥ 0 we have N(C1) ⊆ Dα and since Dα ⊂ E, we can choose K = E.Then by the Kato numerical range mapping theorem (Proposition 2.2) we get:N(f(C1) = F (t/2)2) ⊆ E ′ ⊆ Dα . (3.7)Similarly, take the contraction C2 := F (t/4)2. Since (3.7) is valid for any t ≥ 0,it is true for t 7→ t/2. Then by definition of K one has N(F (t/4)2) ⊆ Dα ⊆ K.Now again the Proposition 2.2 implies:N(f(C2) = F (t/4)4) ⊆ E ′ ⊆ Dα . (3.8)Therefore, we obtain N(Fb(t/2n)2n) ⊆ Dα, for any n ∈ N. By Lemma 2.2 thisyieldslimn→∞(u, (I + t A/2n)−2nu) = (u, e−t Au) ∈ Dα ,for any unit vector u ∈ H. Therefore, the numerical ranges of the contractionsemigroup N(e−t A) ⊆ Dα for all t ≥ 0, if it is generated by m-sectorial operatorwith the semi-angle α ∈ [0, pi/4] and with the vertex at z = 0. ¤4 Corollaries and Applications1. Notice that Definition 2.2 of quasi-sectorial contractions C is quite restric-tive comparing to the notion of general contractions, which demands onlyN(C) ⊆ D1. For the latter case one has a well-known Chernoff lemma [5]:‖(Cn − en(C−I))u‖ ≤ n1/2‖(C − I)u‖ , u ∈ H , n ∈ N , (4.1)7which is not even a convergent bound. For quasi-sectorial contractions we canobtain a much stronger estimate [4]:∥∥∥Cn − en(C−I)∥∥∥ ≤ M n−1/3 , n ∈ N , (4.2)convergent to zero in the uniform topology when n →∞. Notice that the rateof convergence n−1/3 obtained in [4] with help of the Poisson representationand the Tchebychev inequality is not optimal. In [9], [2] and [1] this estimatewas improved up to the optimal rate O(n−1), 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-sectorialcontractions:Proposition 4.1 If C is a quasi-sectorial contraction on a Hilbert space Hwith semi-angle 0 ≤ α < pi/2, i.e. the numerical range N(C) is a subset of thedomain Dα, then‖Cn(I − C)‖ ≤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≥0 be a family of uniformly quasi-sectorial con-tractions on a Hilbert space H, i.e. such that there exists 0 < α < pi/2 andN(Φ(s)) ⊆ Dα, for all s ≥ 0. LetX(s) := (I − Φ(s))/s ,and let X0 be a closed operator with non-empty resolvent set, defined in aclosed subspace H0 ⊆ H. Then the family {X(s)}s>0 converges, when s → +0,in the uniform resolvent sense to the operator X0 if and only iflimn→∞∥∥∥Φ(t/n)n − e−tX0P0∥∥∥ = 0 , for t > 0 . (4.4)Here P0 denotes the orthogonal projection onto the subspace H0.3. We conclude by application of Theorem 2.1 and Proposition 4.1 to the Eulerformula [4], [9], [2].8Proposition 4.3 If A is an m-sectorial operator in a Hilbert space H, withsemi-angle α ∈ [0, pi/4] and with vertex at z = 0, thenlimn→∞∥∥∥(I + tA/n)−n − e−tA∥∥∥ = 0, t ∈ Spi/2−α.Moreover, uniformly in t ≥ t0 > 0 one has the error estimate:∥∥∥(I + tA/n)−n − e−tA∥∥∥ ≤ O(n−1), n ∈ N .AcknowledgementsI 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.References[1] V. Bentkus and V. Paulauskas, Optimal error estimates in operator-normapproximations of semigroups. Lett. Math. Phys. 68 (2004), 131–138.[2] V. Cachia, Euler’s exponential formula for semigroups, Semigroup Forum 68(2004), 1–24.[3] V. Cachia, H. Neidhardt, V.A. Zagrebnov, Comments on the Trotter productformula error-bound estimates for nonself-adjoint semigroups, Integral Equat.Oper. Theory 42 (2002), 425–448.[4] V. Cachia, V.A. Zagrebnov, Operator-norm approximation of semigroups byquasi-sectorial contractions, J. Funct. Anal. 180 (2001), 176–194.[5] P.R. Chernoff, Note on product formulas for operator semigroups. J. Funct.Anal. 2 (1968) 238–242.[6] P.R. Chernoff, Product formulas, nonlinear semigroups, and addition ofunbounded operators. Memoirs of the American Mathematical Society, No.140. American Mathematical Society, Providence, R. I., 1974.[7] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin,1966.[8] T. Kato, Some mapping theorems for the numerical range, Proc. Japan Acad.41 (1966), 652–655.[9] V. Paulauskas, On operator-norm approximation of some semigroups by quasi-sectorial operators, J. Funct. Anal. 207 (2004), 58–67.9[10] M. Uchiyama, Numerical ranges of elements of involutive Banach algebras andcommutativity, Arch.Math. 69 (1997), 313–318.[11] V.A. Zagrebnov, Topics in the theory of Gibbs semigroups. Leuven Notes inMathematical and Theoretical Physics. Series A: Mathematical Physics, 10.Leuven University Press, Leuven, 2003.10-1 1-sinΓΓaββrαα1/2DBa**a*i-i0AΣr-sinFig. 1. Illustration of the set Dα(= Σa∗ shaded domain) with boundary ∂Dα = Γa∗ ,where a∗ = sinα, as well as of our choice of the contour Γr in the resolvent set ρ(C),where r = sinβ > a∗. The contour Γr consists of two segments of tangent straightlines (1, A) and (1, B) and the arc (A,B) of radius r. The dotted circle ∂Dr=1/2corresponds to the set of tangent points for different values of α ∈ [0, pi/2].11

Quasi-sectorial contractions

Journal of Functional Analysis 254 (2008) 2503–2511
www.elsevier.com/locate/jfa
Note
Quasi-sectorial contractions
Valentin A. Zagrebnov a,b,∗
a Université de la Méditerranée (Aix-Marseille II), France
b Centre de Physique Théorique – UMR 6207, Luminy-Case 907, 13288 Marseille Cedex 9, France
Received 7 October 2007; accepted 18 October 2007
Available online 4 December 2007
Communicated by L. Gross
Abstract
We revise the notion of the quasi-sectorial contractions. Our main theorem establishes 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 the theory of operator-norm approximations of strongly continuous semigroups.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Operator numerical range; m-Sectorial generators; Contraction semigroups; Quasi-sectorial contractions;
Holomorphic semigroups; Semigroup operator-norm approximations
1. Sectorial operators
Let H be a separable Hilbert space and let T be a densely defined linear operator with domain
dom(T ) ⊂ H.
Definition 1.1. The set of complex numbers
N(T ) := {(u,T u) ∈ C: u ∈ dom(T ), ‖u‖ = 1},
is called the numerical range of the operator T .
* Address for correspondence: Centre de Physique Théorique – UMR 6207, Luminy-Case 907, 13288 Marseille
Cedex 9, France.
E-mail address: valentin.zagrebnov@cpt.univ-mrs.fr.0022-1236/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2007.10.010
2504 V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511Remark 1.1.
(a) It is known that the set N(T ) is convex (the Toeplitz–Hausdorff theorem), and in general is
neither open nor closed, even for a closed operator T .
(b) Let Δ := C \ N(T ) be complement of the numerical range closure in the complex plane.
Then Δ is a connected open set except the special case, when N(T ) is a strip bounded by
two parallel straight lines.
Below we use some important properties of this set, see e.g. [7, Chapter V], or [11, Chap-
ter 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 complex number z /∈ N(T ), the operator
T − zI is injective. Moreover, it has a closed range ran(T − zI) and a constant deficiency
def(T − zI) in each of connected component of C \N(T ).
(ii) If def(T − zI) = 0 for z /∈ N(T ), then Δ is a subset of the resolvent set ρ(T ) of the operator
T and
∥∥(T − zI)−1∥∥ 1
dist(z,N(T ))
. (1)
(iii) If dom(T ) is dense and N(T ) = C, then T is closable, hence the adjoint operator T ∗ is also
densely defined.
Corollary 1.1. For a bounded operator T ∈ L(H) the spectrum σ(T ) is a subset of N(T ).
For unbounded operator T the relation between spectrum and numerical range is more com-
plicated. For example, it may very well happen that σ(T ) is not contained in N(T ), but for
a closed operator T the essential spectrum σess(T ) is always a subset of N(T ). The condition
def(T − zI) = 0, z /∈ N(T ) in Proposition 1.1(ii) serves to ensure that for those unbounded
operators one gets
σ(T ) ⊂ N(T ), (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/2) and a vertex at z = 0
if
N(T ) ⊆ Sα :=
{
z ∈ C: |arg z| α}.
If, in addition, T is closed and there is z ∈ C \ Sα such that it belongs to the resolvent 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/2) and the vertex at z = 0.
Then it is obvious that numerical ranges of the operators aT and Tb := T + b belong to the same
V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511 2505sector Sα for any non-negative parameters a, b  0. In fact N(Tb) ⊆ Sα + b, i.e. the operator Tb
has the vertex at z = b.
Some of important properties of the m-sectorial operators are summarized by the following.
Proposition 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/2−α}, i.e. the closure: S˜β = Sβ ;
(ii) is a contraction, i.e. N(U(ζ )) is a subset of the unit disc Dr=1 := {z ∈ C: |z|  1} for
{ζ ∈ S˜π/2−α}.
2. Quasi-sectorial contractions and the main theorem
The notion of the quasi-sectorial contractions was introduced in [3] to study the operator-
norm approximations of semigroups. In paper [4] this class of contractions appeared in analysis
of the operator-norm error bound estimate of the exponential Trotter product formula for the
case of accretive perturbations. Further applications of these contractions which, in particular,
improve the rate of convergence estimate of [3] for the Euler formula, one can find in [2,9]
and [1].
Definition 2.1. For α ∈ [0,π/2) we define in the complex plane C a closed domain:
Dα :=
{
z ∈ C: |z| sinα}∪ {z ∈ C: ∣∣arg(1 − z)∣∣ α and |z − 1| cosα}.
This is a convex subset of the unit disc Dr=1, with “angle” (in contrast to tangent) touching of
its boundary ∂Dr=1 at only one point z = 1, see Fig. 1. It is evident that Dα ⊂ Dβ>α .
Definition 2.2. (Quasi-Sectorial Contractions, see [3].) A contraction C on the Hilbert space
H is called quasi-sectorial with semi-angle α ∈ [0,π/2) with respect to the vertex at z = 1, if
N(C) ⊆ Dα .
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 = 0 and α = π/2 correspond, respectively, to
non-negative (self-adjoint) and to general contraction.
The resolvent of an m-sectorial operator A, with semi-angle α ∈ (0,π/4] and vertex at z = 0,
gives the first non-trivial (and for us a key) example of a quasi-sectorial contraction.
Proposition 2.1. Let A be m-sectorial operator with semi-angle α ∈ [0,π/4] and vertex at z = 0.
Then {F(t) := (I + tA)−1}t0 is a family of quasi-sectorial contractions which numerical ranges
N(F (t)) ⊆ Dα for all t  0.
Proof. First, by virtue of Proposition 1.1(ii) we obtain the estimate
∥∥F(t)∥∥ 1
t dist(1/t, −Sα) = 1, (3)
which implies that operators {F(t)}t0 are contractions with numerical ranges N(F (t)) ⊆ Dr=1.
2506 V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511Fig. 1. Illustration of the set Dα (= 
a∗ shaded domain) with boundary ∂Dα = Γa∗ , where a∗ = sinα, as well as of our
choice of the contour Γr in the resolvent set ρ(C), where r = sinβ > a∗ . The contour Γr consists of two segments of
tangent straight lines (1,A) and (1,B) and the arc (A,B) of radius r . The dotted circle ∂Dr=1/2 corresponds to the set
of tangent points for different values of α ∈ [0,π/2].
Next, by Remark 1.2 for all u ∈ H one gets (u,F (t)u) = (vt , vt ) + t (Avt , vt ) ∈ Sα , where
vt := F(t)u, i.e. for any t  0 the numerical range N(F (t)) ⊆ Sα . Similarly, one finds that
(u, (I − F(t))u) = t (v,Av)+ t2(Av,Av) ∈ Sα , i.e., N(I − F(t)) ⊆ Sα . Therefore, for all t  0
we obtain:
N
(
F(t)
)⊆ (Sα ∩ (1 − Sα))⊂ Dr=1. (4)
Moreover, since α  π/4, by Definition 2.1 we get (Sα ∩ (1 − Sα)) ⊂ Dα , i.e. for these values
of α the operators {F(t)}t0 are quasi-sectorial contractions with numerical ranges in Dα . 
Now we are in position to prove the main theorem establishing a relation between quasi-
sectorial contraction semigroups and a certain class of m-sectorial generators.
Theorem 2.1. Let A be an m-sectorial operator with semi-angle α ∈ [0,π/4] and with ver-
tex at z = 0. Then {e−tA}t0 is a quasi-sectorial contraction semigroup with numerical ranges
N(e−tA) ⊆ Dα for all t  0.
The proof of the theorem is based on a series of lemmata and on the numerical range mapping
theorem by Kato [8] (see also an important comment about this theorem in [10]).
Proposition 2.2. (See [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 inverse function f−1 : E′ →
E ⊇ K , where K is a convex kernel of E, i.e., a subset of 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′.
V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511 2507Notice that for a convex set E the corresponding convex kernel K = E.
Lemma 2.1. Let fn(z) = zn be complex functions, for z ∈ C and n ∈ N. Then the sets fn(Dα)
are convex and domains fn(Dα) ⊆ Dα for any n ∈ N, if α  π/4.
Lemma 2.2. (Euler formula). Let A be an m-sectorial operator. Then for t  0 one gets the
strong limit
s- lim
n→∞
(
F(t/n)
)n = e−tA. (5)
The next section is reserved for the proofs. They refine and modify some lines of reasonings of
the paper [3]. This concerns, in particular, a corrected proofs of Proposition 2.1 and Theorem 2.1
(cf. [3, Theorem 2.1]), as well as reformulations and proofs of Proposition 2.2 and Lemma 2.1.
3. Proofs
Proof of Lemma 2.1. Let {z: |z|  sinα} ⊂ Dα , then one gets |zn|  sinα. Therefore, for the
mappings fn : z → zn one obtains fn(z) ∈ Dα for any n 1.
Thus, it rests to check the same property only for images fn(Gα), n 1 of the sub-domain:
Gα :=
{
z:
∣∣arg(z)∣∣< (π/2 − α)}∩ {z: ∣∣arg(z + 1)∣∣> (π − α)}⊂ Dα, (6)
see Definition 2.1 and Fig. 1.
For 0 t  cosα, two segments of tangent straight intervals:
{
ζ±(t) = 1 + tei(π∓α)
}
0tcosα ⊂ ∂Dα,
are correspondingly upper ζ+(t) and lower ζ−(t) = ζ+(t) non-arc parts of the total boundary
∂Dα ; they also coincide with a part of the boundary ∂Gα connected to the vertex z = 1.
Now we proceed by induction. Let n = 1. Then one obviously obtain fn=1(Dα) = Dα . For
n = 2 the boundary ∂f2(Gα) of domain f2(Gα) is a union Γ2(α)∪ Γ2(α) of the contour
Γ2(α) :=
{
f2
(
ζ+(t)
)}
0tcosα ∪
{
z: |z| sin2 α, arg(z) = (π − 2α)}
and its conjugate Γ2(α). Since arg(∂tf2(ζ+(t))) (π − α) for all 0 t  cosα, the contour{
f2
(
ζ+(t)
)}
0tcosα ⊆
{
z:
∣∣arg(z + 1)∣∣> (π − α)},
see (6). The same is obviously true for the image of the lower branch ζ−(t). If α  π/4, one gets
sup
0tcosα
Im
(
f2
(
ζ+(t)
))= Im(f2(ζ+(t∗ = (2 cosα)−1)))
= 1 tanα < sinα cosα, (7)
2
2508 V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511where t∗ = (2 cosα)−1  cosα, and
0 Re
(
f2
(
ζ+(t)
))
− sin2 α cos 2α − sinα.
Therefore, {f2(ζ+(t))}0tcosα ⊆ Dα . Since the same is also true for the image of the lower
branch ζ−(t), we obtain f2(Gα) ⊂ Dα and by consequence fn=2(Dα) = {w = z · z: z ∈ Dα, z ∈
fn=1(Dα)} ⊂ Dα, for α  π/4.
Now let n > 2 and suppose that fn(Dα) ⊂ Dα . Then the image of the (n+ 1)-order mapping
of domain Dα is:
fn+1(Dα) =
{
w = z · zn: z ∈ Dα, zn ∈ fn(Dα)
}
,
and since fn(Dα) ⊂ Dα , we obtain fn+1(Dα) ⊂ Dα by the same reasoning as for n = 2. 
Remark 3.1. Let φ(t) := arg(ζ+(t)). Then cot(α + φ(t)) = (cosα − t)/ sinα and
sup
0tcosα
Im
(
fn
(
ζ+(t)
))

(
1 − 2t∗n cosα +
(
t∗n
)2)n/2 (8)
for sin(nφ(t∗n )) = 1. In the limit n → ∞ this implies that φ(t∗n ) = π/2n + o(n−1), t∗n =
π/(2n sinα)+ o(n−1) and
lim
n→∞ sup0tcosα
Im
(
fn
(
ζ+(t)
))
 exp
(
−1
2
π cotα
)
<
1
2
tanα. (9)
By the same reasoning one gets the estimates similar to (8) and (9) for ζ−(t)). Hence,
|Im(fn(ζ±(t)))| < Im(fn=1(ζ+(t))) < sinα cosα, cf. (7).
Notice that in spite of the arc-part of the contour ∂Dα shrinks in the limit n → ∞ to zero, we
obtain
lim
n→∞ sup0tcosα
Re
(
fn
(
ζ+(t)
))= − exp(−π cotα), (10)
for the left extreme point of the projection on the real axis (cos(nφ(t∗n )) = −1) of the image
fn(Dα). Since exp(−π cotα) < sinα, for α  π/4, the arguments (9) and (10) bolster the con-
clusion of Lemma 2.1.
Proof of Lemma 2.2. By (3) we have for λ > 0∥∥(λI +A)−1∥∥< λ−1, (11)
and since A is m-sectorial, we also get that (−∞,0) ⊂ ρ(A). Then the Hille–Yosida theory
ensures the existence of the contraction semigroup {e−tA}t0, and the standards arguments (see
e.g. [7, Chapter V], or [11, Chapter 1.1]) yield the convergence of the Euler formula (5) in the
strong topology. 
Proof of Theorem 2.1. Take f (z) = z2 and the compact convex set E′ := f (Dα) ⊆ Dα , see
Lemma 2.1. Since the set E := f−1(E′) = Dα ∪ (−Dα) is convex, its convex kernel K exists
V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511 2509and K = E. Then by Proposition 2.2 we obtain that N(f (C)) ⊆ E′ ⊆ Dα , if the numerical range
N(C) ⊆ K .
Let contraction C1 := (I + tA/2)−1 = F(t/2). Since by Proposition 2.1 for any t  0 we
have N(C1) ⊆ Dα and since Dα ⊂ E, we can choose K = E. Then by the Kato numerical range
mapping theorem (Proposition 2.2) we get
N
(
f (C1) = F(t/2)2
)⊆ E′ ⊆ Dα. (12)
Similarly, take the contraction C2 := F(t/4)2. Since (12) is valid for any t  0, it is true for
t → t/2. Then by definition of K one has N(F (t/4)2) ⊆ Dα ⊆ K . Now again Proposition 2.2
implies
N
(
f (C2) = F(t/4)4
)⊆ E′ ⊆ Dα. (13)
Therefore, we obtain N(Fb(t/2n)2
n
) ⊆ Dα , for any n ∈ N. By Lemma 2.2 this yields
lim
n→∞
(
u,
(
I + tA/2n)−2nu)= (u, e−tAu) ∈ Dα,
for any unit vector u ∈ H. Therefore, the numerical ranges of the contraction semigroup
N(e−tA) ⊆ Dα for all t  0, if it is generated by m-sectorial operator with the semi-angle
α ∈ [0,π/4] and with the vertex at z = 0. 
4. Corollaries and applications
1. Notice that Definition 2.2 of quasi-sectorial contractions C is quite restrictive comparing
to the notion of general contractions, which demands only N(C) ⊆ D1. For the latter case one
has a well-known Chernoff lemma [5]:∥∥(Cn − en(C−I ))u∥∥ n1/2∥∥(C − I )u∥∥, u ∈ H, n ∈ N, (14)
which is not even a convergent bound. For quasi-sectorial contractions we can obtain a much
stronger estimate [3]: ∥∥Cn − en(C−I )∥∥Mn−1/3, n ∈ N, (15)
convergent to zero in the uniform topology when n → ∞. Notice that the rate of convergence
n−1/3 obtained in [3] with help of the Poisson representation and the Tchebychev inequality is
not optimal. In [2,9] and [1] this estimate was improved up to the optimal rate O(n−1), which
one can easily verify for a particular case of self-adjoint contractions (i.e. α = 0) with help of the
spectral representation.
The inequality (15) and its further improvements are based on the following important 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
0 α < π/2, i.e. the numerical range N(C) is a subset of the domain Dα , then
∥∥Cn(I −C)∥∥ K
n+ 1 , n ∈ N. (16)
2510 V.A. Zagrebnov / Journal of Functional Analysis 254 (2008) 2503–2511For the proof see [3, Lemma 3.1].
2. Another application of quasi-sectorial contractions generalizes the Chernoff semigroup ap-
proximation theory [5,6] to the operator-norm approximations [3].
Proposition 4.2. Let {Φ(s)}s0 be a family of uniformly quasi-sectorial contractions on a Hilbert
space H, i.e. such that there exists 0 < α < π/2 and N(Φ(s)) ⊆ Dα , for all s  0. Let
X(s) := (I −Φ(s))/s,
and let X0 be a closed operator with non-empty resolvent set, defined in a closed subspace
H0 ⊆ H. Then the family {X(s)}s>0 converges, when s → +0, in the uniform resolvent sense to
the operator X0 if and only if
lim
n→∞
∥∥Φ(t/n)n − e−tX0P0∥∥= 0, for t > 0. (17)
Here P0 denotes the orthogonal projection onto the subspace H0.
3. We conclude by application of Theorem 2.1 and Proposition 4.1 to the Euler formula [2,3,
9].
Proposition 4.3. If A is an m-sectorial operator in a Hilbert space H, with semi-angle α ∈
[0,π/4] and with vertex at z = 0, then
lim
n→∞
∥∥(I + tA/n)−n − e−tA∥∥= 0, t ∈ Sπ/2−α.
Moreover, uniformly in t  t0 > 0 one has the error estimate:∥∥(I + tA/n)−n − e−tA∥∥O(n−1), n ∈ N.
Acknowledgments
I would like to thank Professor Mitsuru Uchiyama for a useful remark indicating a flaw in
our arguments in Section 2 of [3], revision of this part of the paper [3] is done in the present
manuscript. I also thankful to Vincent Cachia and Hagen Neidhardt for a pleasant collaboration.
References
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[2] V. Cachia, Euler’s exponential formula for semigroups, Semigroup Forum 68 (2004) 1–24.
[3] V. Cachia, V.A. Zagrebnov, Operator-norm approximation of semigroups by quasi-sectorial contractions, J. Funct.
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[7] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.
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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 the theory of operator-norm approximations of strongly continuous semigroups. Key words: Operator numerical range; m-sectorial generators; contraction semigroups; quasi-sectorial contractions; holomorphic semigroups; semigroup operator-norm approximations

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Quasi-Sectorial Contractions

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