Results 1  10
of
15
the H ∞ calculus and sums of closed operators
 LeM] [Mar] [McI] [PT] [She] [Ste] [ST] [Wei] C. Le
"... Abstract. We develop a very general operatorvalued 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 DoreVenni type on sums of commuting sect ..."
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Cited by 27 (4 self)
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Abstract. We develop a very general operatorvalued 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 DoreVenni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the Rboundedness 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.
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Hardy spaces and divergence operators on strongly Lipschitz domain
 of R n , J. Funct. Anal
"... Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under su ..."
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Cited by 10 (1 self)
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Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 z (Ω). A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.
Schrödinger operators with complexvalued potentials and no resonances
 Duke Math Jour
"... Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophas ..."
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Cited by 10 (6 self)
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Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on R d. In odd dimensions d ≥ 3 we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time. 1.
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 8 (3 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
WEIGHTED ADMISSIBILITY AND WELLPOSEDNESS OF LINEAR SYSTEMS IN BANACH SPACES
, 2006
"... Abstract. We study linear control systems in infinite–dimensional Banach spaces governed by analytic semigroups. For p ∈ [1, ∞] and α ∈ R we introduce the notion of L p –admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first ..."
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Cited by 6 (2 self)
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Abstract. We study linear control systems in infinite–dimensional Banach spaces governed by analytic semigroups. For p ∈ [1, ∞] and α ∈ R we introduce the notion of L p –admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first named author and Le Merdy [12] we give conditions under which L p –admissibility of type α is characterised by boundedness conditions which are similar to those in the well–known Weiss conjecture. We also study L p –wellposedness of type α for the full system. Here we use recent ideas due to Pruess and Simonett. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [4] to non–Hilbertian settings and to p ̸ = 2.
ON A QUADRATIC ESTIMATE RELATED TO THE KATO CONJECTURE AND BOUNDARY VALUE PROBLEMS
, 810
"... Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a ..."
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Cited by 3 (3 self)
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Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms. MSC classes: 35J25, 35J55, 47N20, 47F05, 42B25 Keywords: LittlewoodPaley estimate, functional calculus, boundary value problems, second order elliptic equations and systems, square root problem 1.
ON THE CARLESON MEASURE CRITERION IN LINEAR SYSTEMS THEORY
, 804
"... Abstract. In Ho, Russell [15], and Weiss [33], a Carleson measure criterion for admissibility of onedimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation X = ℓ2 and L 2 –admissibility to the more general situation of L p –a ..."
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Cited by 1 (1 self)
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Abstract. In Ho, Russell [15], and Weiss [33], a Carleson measure criterion for admissibility of onedimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation X = ℓ2 and L 2 –admissibility to the more general situation of L p –admissibility on ℓq–spaces. In case of analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to result of L p –admissibility for reciprocal systems in the sense of Curtain [5].
The Stieltjes Convolution and a Functional Calculus for Nonnegative Operators Abstract
"... In this paper we present an approach to the multidimensional distributional Stieltjes transform that allows us to define a convolution operation on our classes of Stieltjestransformable distributions. As an application, we develop a powerful and intuitive functional calculus for (possibly multiple) ..."
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In this paper we present an approach to the multidimensional distributional Stieltjes transform that allows us to define a convolution operation on our classes of Stieltjestransformable distributions. As an application, we develop a powerful and intuitive functional calculus for (possibly multiple) nonnegative operators with which one can easily prove a variety of operator equations, even for noncommuting operators and on noncomplex Banach spaces. Two representation theorems that identify our classes of distributions as finite sums of derivatives of functions that fulfill certain estimates are essential throughout this paper.
Quadratic Estimates and . . .
, 2004
"... We prove quadratic estimates for complex perturbations of Diractype operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L∞ changes in the metric ..."
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We prove quadratic estimates for complex perturbations of Diractype operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on