## the H ∞ -calculus and sums of closed operators

Venue: | LeM] [Mar] [McI] [PT] [She] [Ste] [ST] [Wei] C. Le |

Citations: | 27 - 4 self |

### BibTeX

@ARTICLE{Kalton_theh,

author = {N. J. Kalton and L. Weis},

title = {the H ∞ -calculus and sums of closed operators},

journal = {LeM] [Mar] [McI] [PT] [She] [Ste] [ST] [Wei] C. Le},

year = {},

pages = {137--156}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We develop a very general operator-valued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting. 1.

### Citations

89 | Equations of Evolution - Tanabe - 1979 |

85 |
Factorization of linear operators and geometry of Banach spaces
- Pisier
- 1986
(Show Context)
Citation Context ...We recall that a GT-space is a Banach space X so that the Grothendieck theorem is valid, i.e. if T : X → ℓ2 is bounded then T is absolutely summing and for some C independent of T, π1(T) ≤ C‖T ‖; see =-=[38]-=- for a full discussion. Examples of such spaces are L1−spaces, their quotients by reflexive spaces and L1/H1 ( [7], [9] and [38]). Proposition 7.1. Suppose X is a GT-space of cotype 2 and that A is a ... |

44 |
Absolutely summing operators in Lp-spaces and their applications
- Lindenstrauss, Pelczyński
- 1968
(Show Context)
Citation Context ...what simpler than preceding approachs (even for Hilbert spaces). Finally in Section 7 we use this observation to show how classical results on unconditional bases due to Lindenstrauss and Pe̷lczyński =-=[34]-=- can be recast as results on operators with an H ∞ −calculus on L1 and C(K)−spaces. In these cases we get very strong conclusions, but they are mitigated by the observation that there are in practice ... |

39 |
Holomorphic semi-groups and the geometry of Banach spaces
- Pisier
- 1982
(Show Context)
Citation Context ... really non-trivial part of the Lemma is the last sentence and this follows easily from Pisier’s characterization of spaces with nontrivial type as those in which the Rademacher projection is bounded =-=[37]-=-. We shall also need some related Banach space concepts. Suppose (ǫk) ∞ k=1 and (ηk) ∞ k=1 are two mutually independent sequences of Rademachers. We say that X has property (α) (see [36] and [28]) if ... |

37 |
Operator-valued Fourier multiplier theorems and maximal Lp-regularity
- Weis
- 2001
(Show Context)
Citation Context ...on 3 we recall the notion of Rademacher-boundedness (or R-boundedness) of families of operators. This implicitly goes back to work of Bourgain [6] and has recently been studied in [4], [10], [11] and =-=[41]-=- in connection with vector-valued multiplier theorems. We also introduce some weaker notions and study their relationship to certain Banach space properties of the underlying space. Using these ideas ... |

35 |
Some remarks on Banach spaces in which martingale difference sequences are unconditional,” Arkiv für
- Bourgain
- 1983
(Show Context)
Citation Context ...rial operators and an operator-valued extension. In Section 3 we recall the notion of Rademacher-boundedness (or R-boundedness) of families of operators. This implicitly goes back to work of Bourgain =-=[6]-=- and has recently been studied in [4], [10], [11] and [41] in connection with vector-valued multiplier theorems. We also introduce some weaker notions and study their relationship to certain Banach sp... |

29 |
On the closedness of the sum of two closed operators
- Dore, Venni
- 1987
(Show Context)
Citation Context ...n the boundedness condition (1.1) is replaced by an Rboundedness condition. Using this in Theorem 6.3 we prove a general result on sums which can be regarded as an extension of the Dore-Venni Theorem =-=[18]-=-. We show that if A, B are commuting sectorial operators such that A has an H ∞ (Σσ)−calculus and B is R-sectorial of type σ ′ when σ + σ ′ < π then A + B (with domain D(A) ∩ D(B)) is closed. One prac... |

27 | Vector-valued singular integrals and the H 1 − BMO duality. Probability theory and harmonic - Bourgain - 1986 |

25 |
Sommes d’opérateurs linéaires et équations différentielles opérationnelles
- Prato, Grisvard
- 1975
(Show Context)
Citation Context ...ting sectorial operators on a Banach space X with ω(A)+ω(B) < π then one can show that the closure A + B of A+B with D(A+ B) = D(A) ∩ D(B) is a sectorial operator with ω(A + B) ≤ max(ω(A), ω(B)) (see =-=[15]-=-). However, in many applications one needs to show that A + B is already closed on D(A) ∩ D(B). We now give a criterion for this. Theorem 6.3. Suppose A and B are commuting sectorial operators such th... |

23 |
Some results on Banach spaces without local unconditional structure
- Pisier
- 1978
(Show Context)
Citation Context ...on is bounded [37]. We shall also need some related Banach space concepts. Suppose (ǫk) ∞ k=1 and (ηk) ∞ k=1 are two mutually independent sequences of Rademachers. We say that X has property (α) (see =-=[36]-=- and [28]) if there is a constant C so that for any (xjk) n j,k=1 ⊂ X and any (αjk) n j,k=1 ⊂ C we have n∑ n∑ n∑ n∑ (E‖ ǫjηkxjk‖ 2 ) 1 (3.4) 2. j=1 k=1 αjkǫjηkxjk‖ 2 ) 1 2 ≤ C max j,k |αjk|(E‖ k=1 j=1... |

21 |
Schauder decomposition and multiplier theorems
- Clement, Pagter, et al.
- 2000
(Show Context)
Citation Context ...nsion. In Section 3 we recall the notion of Rademacher-boundedness (or R-boundedness) of families of operators. This implicitly goes back to work of Bourgain [6] and has recently been studied in [4], =-=[10]-=-, [11] and [41] in connection with vector-valued multiplier theorems. We also introduce some weaker notions and study their relationship to certain Banach space properties of the underlying space. Usi... |

20 |
New Banach space properties of the disc algebra and H
- Bourgain
(Show Context)
Citation Context ...ded then T is absolutely summing and for some C independent of T, π1(T) ≤ C‖T ‖; see [38] for a full discussion. Examples of such spaces are L1−spaces, their quotients by reflexive spaces and L1/H1 ( =-=[7]-=-, [9] and [38]). Proposition 7.1. Suppose X is a GT-space of cotype 2 and that A is a H∞−sectorial operator on X. Then if ωH(A) < ν and 0 < s < 1 there is a constant C so that if x ∈ X, (7.1) C −1 ∫ ‖... |

17 |
Operators which have an H ∞ functional calculus, in Miniconference on operator theory and partial differential equations
- McIntosh
- 1986
(Show Context)
Citation Context ...r functions f which are bounded and analytic on the sector Σσ = {ζ : | argζ| < σ} and which obey a condition of the type |f(ζ)| ≤ C(|ζ|/(1 + |ζ| 2 ) ǫ ) for some ǫ > 0. This is described in detail in =-=[35]-=- and [28] and we give a somewhat different approach in Section 2 below. If we then have an estimate ‖f(A)‖ ≤ C‖f‖H ∞ (Σσ) it is possible to extend the definition of f(A) to any f ∈ H ∞ (Σσ) and we say... |

16 |
Spectral decompositions and harmonic analysis on UMD Banach spaces
- Berkson, Gillespie
- 1994
(Show Context)
Citation Context ... extension. In Section 3 we recall the notion of Rademacher-boundedness (or R-boundedness) of families of operators. This implicitly goes back to work of Bourgain [6] and has recently been studied in =-=[4]-=-, [10], [11] and [41] in connection with vector-valued multiplier theorems. We also introduce some weaker notions and study their relationship to certain Banach space properties of the underlying spac... |

16 |
G.: Bounded linear operators between C ∗ -algebras
- Haagerup, Pisier
- 1993
(Show Context)
Citation Context ...roof using Rademacher in place of Steinhaus variables in the above argument is possible for this case. Thus the Schatten classes Cp have property (∆) as long as 1 < p < ∞. However Haagerup and Pisier =-=[22]-=- show that C1 (which has cotype 2) fails analytic UMD and their argument actually shows it fails property (∆). This implies that C(K)-spaces of infinite dimension also fail (∆) since C1 is finitely re... |

13 | Heat kernels and maximal L p -L q estimates for parabolic evolution equations - Hieber, Prüss - 1997 |

13 |
Functional calculi for linear operators in vector-valued L p -spaces via the transference principle, Adv. Differential Equations 3
- Hieber, Prüss
- 1998
(Show Context)
Citation Context ... that f has an H ∞ (Σσ)−calculus. It is, by now, well-known that many systems of parabolic differential operators, Schrödinger operators and pseudodifferential operators do have an H ∞ −calculus (cf. =-=[24]-=-, [3], [2] and [19]) and this has proved useful in applications. Of particular importance are two closely related problems: 1991 Mathematics Subject Classification. Primary: 47A60, 47D06. The first au... |

11 | Maximal regularity and kernel bounds: observations on a theorem by Hieber and - Coulhon, Duong |

9 |
p regularity for abstract differential equations, Functional analysis and related topics
- Dore, L
- 1991
(Show Context)
Citation Context ...mal 2 Lp−regularity. (3) It is shown in [41] that the condition ωR(A) < π actually characterizes 2 Lp-maximal regularity. The operator approach to maximal regularity has a long history, see e.g. [15],=-=[16]-=-, [21] and [42]. 7. L1-spaces and C(K)-spaces We recall that a GT-space is a Banach space X so that the Grothendieck theorem is valid, i.e. if T : X → ℓ2 is bounded then T is absolutely summing and fo... |

9 |
Equations différentielles abstraites, Ann
- Grisvard
- 1969
(Show Context)
Citation Context ...Lp−regularity. (3) It is shown in [41] that the condition ωR(A) < π actually characterizes 2 Lp-maximal regularity. The operator approach to maximal regularity has a long history, see e.g. [15],[16], =-=[21]-=- and [42]. 7. L1-spaces and C(K)-spaces We recall that a GT-space is a Banach space X so that the Grothendieck theorem is valid, i.e. if T : X → ℓ2 is bounded then T is absolutely summing and for some... |

8 |
Martingale transforms and complex uniform convexity
- Bourgain, Davis
(Show Context)
Citation Context ...hen T is absolutely summing and for some C independent of T, π1(T) ≤ C‖T ‖; see [38] for a full discussion. Examples of such spaces are L1−spaces, their quotients by reflexive spaces and L1/H1 ( [7], =-=[9]-=- and [38]). Proposition 7.1. Suppose X is a GT-space of cotype 2 and that A is a H∞−sectorial operator on X. Then if ωH(A) < ν and 0 < s < 1 there is a constant C so that if x ∈ X, (7.1) C −1 ∫ ‖x‖ ≤ ... |

7 |
Bounded H∞-calculus for elliptic operators
- Amann, Hieber, et al.
- 1994
(Show Context)
Citation Context ... an H ∞ (Σσ)−calculus. It is, by now, well-known that many systems of parabolic differential operators, Schrödinger operators and pseudodifferential operators do have an H ∞ −calculus (cf. [24], [3], =-=[2]-=- and [19]) and this has proved useful in applications. Of particular importance are two closely related problems: 1991 Mathematics Subject Classification. Primary: 47A60, 47D06. The first author ackno... |

7 |
The square root problem of Kato in one dimension and first order elliptic systems
- Auscher, McIntosh, et al.
- 1997
(Show Context)
Citation Context ...f has an H ∞ (Σσ)−calculus. It is, by now, well-known that many systems of parabolic differential operators, Schrödinger operators and pseudodifferential operators do have an H ∞ −calculus (cf. [24], =-=[3]-=-, [2] and [19]) and this has proved useful in applications. Of particular importance are two closely related problems: 1991 Mathematics Subject Classification. Primary: 47A60, 47D06. The first author ... |

7 |
An operator-valued transference principle and maximal regularity on vector-valued Lp-spaces
- Clément, Prüss
- 2001
(Show Context)
Citation Context ...erators on L2(X) such an operatorvalued functional calculus is constructed in [28] Theorem 5.2 and [30]. These are cases when the U-boundedness condition is satisfied automatically. See also [20]. In =-=[11]-=- there are constructions based on transference results which work for generators of bounded c0-groups on UMD-spaces and some other special cases. Proof. As before we consider ϕnF = Fn so that Fn ∈ H(A... |

7 |
Merdy, A joint functional calculus for sectorial operators with commuting resolvents
- Lancien, Lancien, et al.
- 1998
(Show Context)
Citation Context ...ns f which are bounded and analytic on the sector Σσ = {ζ : | argζ| < σ} and which obey a condition of the type |f(ζ)| ≤ C(|ζ|/(1 + |ζ| 2 ) ǫ ) for some ǫ > 0. This is described in detail in [35] and =-=[28]-=- and we give a somewhat different approach in Section 2 below. If we then have an estimate ‖f(A)‖ ≤ C‖f‖H ∞ (Σσ) it is possible to extend the definition of f(A) to any f ∈ H ∞ (Σσ) and we say that f h... |

6 | A solution to the problem of the L p -maximal regularity
- Kalton, Lancien
- 1999
(Show Context)
Citation Context ...hat ωR( Ã) = ωR(A) so the result is a consequence of Theorem 6.3. Remarks. (1) If X has (UMD) and A is H∞−sectorial then ωH(A) < π 2 implies maximal Lp-regularity by Corollary 6.4. (2) It is shown in =-=[25]-=- that any non-Hilbertian Banach space with an unconditional basis admits a sectorial operator A with ω(A) < π but failing maximal 2 Lp−regularity. (3) It is shown in [41] that the condition ωR(A) < π ... |

4 | Merdy, H ∞ functional calculus and applications to maximal regularity - Le - 1999 |

3 | Régularité L p pour les équations d’évolution, Séminaire d’Analyse Fonctionnelle - Coulhon, Lamberton |

3 | Merdy, Counterexamples on L p -maximal regularity - Le - 1999 |

2 |
Banach space operators with a bounded H
- Cowling, Doust, et al.
- 1986
(Show Context)
Citation Context ... < π 2 .16 N. J. KALTON AND L. WEIS Proof. It is shown in [24] that ωH(A) < π and in [42], Section 5, that ωR(A) < . Hence we can apply Proposition 5.1. π 2 We remark that it is an open problem (cf. =-=[14]-=-) whether ωH(A) = ω(A) whenever A is H ∞ −sectorial. The next Theorem gives some results in this direction. Theorem 5.3. Suppose A is an H ∞ −sectorial operator on a Banach space X. Then: 1. If X has ... |

2 | Maximal regularity in L p spaces for an abstract Cauchy problem - Dore |

2 |
Maximal Lp−regularity for second order elliptic operators with uniformly continuous coefficients on domains, preprint
- Kunstmann
(Show Context)
Citation Context ...e σ ′ when σ + σ ′ < π then A + B (with domain D(A) ∩ D(B)) is closed. One practical advantage of our result is that it is easier to check R-sectoriality than the boundedness of imaginary powers (see =-=[27]-=-, [42] and [11]). We also give applications to the joint H ∞ −functional calculus (cf. [30]) and show how Banach space properties such as UMD, analytic UMD and property (α) of Pisier relate to our res... |

2 |
A new approach to maximal Lp-regularity, to appear in
- Weis
- 1998
(Show Context)
Citation Context ...rity. (3) It is shown in [41] that the condition ωR(A) < π actually characterizes 2 Lp-maximal regularity. The operator approach to maximal regularity has a long history, see e.g. [15],[16], [21] and =-=[42]-=-. 7. L1-spaces and C(K)-spaces We recall that a GT-space is a Banach space X so that the Grothendieck theorem is valid, i.e. if T : X → ℓ2 is bounded then T is absolutely summing and for some C indepe... |

1 | Holomorphic functional calculi and sums of commuting operators
- Albrecht, Franks, et al.
- 1998
(Show Context)
Citation Context ...if f ∈ H∞ (Σσ × Σσ ′) then the family {f(z, B) : z ∈ H∞ (Σσ)} is WR-bounded and then apply Theorem 4.4. We can now apply Theorem 5.3 to obtain a result of Lancien, Lancien and Le Merdy [28] (see also =-=[1]-=-). Note that their argument depends on the quite technical discretization developed in [20]. Corollary 6.2. (Lancien, Lancien and Le Merdy [28]) If X has property (A) then if A and B are commuting H ∞... |

1 |
A multiplier characterization of analytic UMD spaces
- Blower
- 1990
(Show Context)
Citation Context ... j∑ j=1 k=1 ǫjηkxjk‖ 2 ) 1 2 ≤ C(E‖ n∑ n∑ j=1 k=1 ǫjηkxjk‖ 2 ) 1 2. It is clear that (∆) is a weaker property than (α). It is in fact shared by all spaces with (UMD) and even analytic UMD. We recall (=-=[5]-=-) that X has analytic UMD if every L1−bounded analytic martingale has unconditional martingale differences. Proposition 3.2. Suppose X has analytic UMD. Then X has property (∆). Proof. Let (˜ǫk) ∞ k=1... |

1 |
Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients
- Duong, McIntosh
- 1996
(Show Context)
Citation Context ...(Σσ)−calculus. It is, by now, well-known that many systems of parabolic differential operators, Schrödinger operators and pseudodifferential operators do have an H ∞ −calculus (cf. [24], [3], [2] and =-=[19]-=-) and this has proved useful in applications. Of particular importance are two closely related problems: 1991 Mathematics Subject Classification. Primary: 47A60, 47D06. The first author acknowledges s... |

1 |
Discrete quadratic estimates and holomorphic functional calculi in Banach spaces
- Franks, McIntosh
- 1998
(Show Context)
Citation Context ...ounded operator B(A + B) −1 . This then is a special case of the problem of constructing a joint functional calculus of A, B. In the case of Hilbert spaces and some related situations it was shown in =-=[20]-=-, [28] and [30] that one can construct an operatorvalued functional calculus associated to an operator with H ∞ −calculus and this permits a solution; however, it was also shown that such an approach ... |

1 | Counterexamples concerning sectorial operators - Lancien - 1998 |

1 |
Merdy, A generalized H ∞ -calculus for operators on subspaces of Lp and applications to maximal regularity
- Lancien, Le
- 1998
(Show Context)
Citation Context ... B(A + B) −1 . This then is a special case of the problem of constructing a joint functional calculus of A, B. In the case of Hilbert spaces and some related situations it was shown in [20], [28] and =-=[30]-=- that one can construct an operatorvalued functional calculus associated to an operator with H ∞ −calculus and this permits a solution; however, it was also shown that such an approach cannot work in ... |

1 |
Sunms of non-commuting sectorial operators
- Strkalj
(Show Context)
Citation Context ...41] Theorem 5.2, [42] and more recently in [11] (e.g. if A is the generator of a strongly continuous group in a UMD-space). (3) An extension to non-commuting sums will be given in a forthcoming paper =-=[39]-=-. Proof. Choose σ, σ ′ with ωH(A) < σ, ωR(B) < σ ′ and σ+σ ′ < π. The function f(w, z) = w(w +z) −1 is in H∞ (Σσ ×Σσ ′) and the set f(w, B) = −wR(−w, B) for w ∈ Σσ is an R-bounded family. Applying The... |