IX Escuela-Taller de Análisis Funcional (2019) Holom orphic functional calculus for sectorial o p e r a t o r s � Héctor Ariza A b s t r a c t : This article is about holomorphic functional calculus on Hilbert spaces. Universitat de València It is an expository paper where we survey the main results in this setting and we [email protected] present some examples. We define the natural functional calculus for sectorial op- erators and give an example of a sectorial operator that does not admit a functional � Lucía Cabezas calculus. Universidad de Almería [email protected] R e s u m e n : Este artículo trata sobre el cálculo funcional holomorfo en espacios de Hilbert. Es un artículo expositorio donde tratamos los resultados principales � Javier Canto y presentamos algunos ejemplos. Definimos el cálculo funcional natural para BCAM – Basque Center for Applied operadores sectoriales y damos un ejemplo de un operador sectorial que no admite Mathematics un cálculo funcional. [email protected] K e y w o r d s : holomorphic functional calculus, sectorial operators, Riesz-Dunford. � Gonzalo Cao Labora M S C 2 0 1 0 : 47A60, 47-01. Universitat Politècnica de Catalunya [email protected] � M aría Luisa C.Godoy Universidad de Granada [email protected] � M aría delM ar Utrera Universidad de Almería [email protected] Course instructor Jorge Betancor Universidad de La Laguna [email protected] Acknowledgem ents: We want to thank Domingo García, Manuel Maestre and Carlos Pérez for the organization of the IX Escuela-Taller de la Red de Análisis Funcional y Aplicaciones, held in March 2019 at Euskal Herriko Unibertsitatea. This workshop gave us the opportunity to work with the ideas exposed in this article. Wealso want to thank our advisor Jorge Betancor, without whom it would not have been possible to complete this work. R e f e r e n c e : ARIZA, Héctor; CABEZAS, Lucía; CANTO, Javier; CAO LABORA, Gonzalo; GODOY, María Luisa C.; UTRERA, María del Mar, and BETANCOR, Jorge. “Holomorphic functional calculus for sectorial operators”. In: TEMat monográficos, 1 (2020): Artículos de las VIII y IX Escuela-Taller de Análisis Funcional, pp. 61-77. ISSN: 2660- 6003. URL: https://temat.es/monograficos/article/view/vol1-p61. cb This work is distributed under a Creative Commons Attribution 4.0 International licence https://creativecommons.org/licenses/by/4.0/ Holomorphic functional calculus 1 . Introduction In this expository paper we develop the main ideas about holomorphic functional calculus for sectorial operators on Hilbert spaces. Of course, we cannot be exhaustive and make a complete presentation of the theory. We present the sketch of some proofs, and the complete demonstration of some results. Our purpose is to show some of the usual procedures and tools in this context. The interested reader can consult, for instance, the references listed at the end. The functional calculus that we study was formalized in the late 80’s and 90’s mainly byMcIntosh[1, 4, 17, 18]. The motivation was in Kato’s square root problem and the operator-method approach to evolution equations of Grisvard and Da Prato. Many examples within the class of operators under consideration can be found among the partial differential operators (elliptic differential operators, Schrödinger operators with singular potentials, Stokes operators...). The purpose of a functional calculus is to give a meaning to 푓(푇), where 푓 is a complex function defined on a subset of ℂ and 푇 is an operator defined on a Hilbert space. For example, in a finite dimensional Hilbert space, we can find exponentials and logarithms ofmatrices through the theory of linear systems of differential equations. These operations can be regarded asa special case of a functional calculus for operators in a finite dimensional setting. Another example is the well-known equality − ( 1 ) Δ푓 = ℱ (|푦| ℱ(푓)) 1 2 where Δ represents the Laplace operator, ℱ denotes the Fourier transform and ℱ− is the inverse of ℱ. , This equality holds provided that 푓 satisfies certain regularity and decay conditions.1 From(1) it follows that, if 푃 is a polynomial, we can compute 푃(Δ) using the following expression: − ( 2 ) 푃(Δ)푓 = ℱ (푃(|푦| )ℱ(푓)). 1 2 Now, if we want to define 푚(Δ) for a general complex function 푚,(2) suggests the following definition: 푚(Δ)푓 = ℱ− (푚(|푦| )ℱ(푓)). 1 2 This is the first step to construct a functional calculus for the Laplace operator. In asecondstep,we need to specify the Hilbert space 퐻 where 푓 belongs to and the class 휉 of admissible functions 푚 in this functional calculus. Actually, (1) can be seen as a spectral representation for the operator Δ. This fact allows to extend these ideas and to define a functional calculus for operators on Hilbert spaces by using spectral representations. The abstract method of defining a functional calculus follows ideas from Haase[8], and it is roughly done in the following way. If 푇 is an operator in a Hilbert space 퐻, we consider a class 휉 of functions defined on the spectrum 휎(푇) of 푇 and a mapping 훷∶ 휉 → 퐵(퐻), where 퐵(퐻) represents the space of bounded and linear operators on 퐻. 훷 is actually a method to assign an operator 푓(푇) defined by 훷(푓). In a first approach, the class 휉 is formed by smooth enough functions. Later on, this class of functions will be enlarged by including less regular functions. In this second step, 푓(푇) may be unbounded if 푇 is not in 퐵(퐻). Since the purpose of functional calculus is to make computations, it is convenient that 휉 is an algebra and 훷 is an algebra homomorphism. Furthermore, the mapping 훷 should be somehow connected to the operator 푇 (note that we write 푓(푇)). One of such relations should be that 훷((휆 − 푧)− ) = 푅(휆 푇), where 휆 ∈ 휌(푇) and − 푅(휆 푇) = (휆퐼 −푇) is the resolvent operator. Haase [8] presented an abstract1 approach to the construction of functional calculus.1 , This, paper is structured as follows. In section 2 we present the holomorphic functional calculus. The definition and the main properties of the sectorial operators on Hilbert spaces are presented insection 3. The holomorphic function algebras, basic for functional calculus, are studied insection 4. Section 5 is focused on the definitions and properties of the holomorphic functional calculus for sectorial operators. Finally, in section 6 we discuss an example: a sectorial operator in a separable Hilbert space that does not have a bounded holomorphic functional calculus. 62 https://temat.es/monograficos Ariza et al. The theory developed in this paper is concerned with operators in Hilbert spaces and holomorphic functions. The standard definitions and properties about these topics that are used in the sequelcanbe found in the monographs of Rudin [19, 20]. A detailed study about the holomorphic functional calculus for sectorial operators appears, for instance, in [7, 14, 15, 24], where most of the results in this paper are included. 2 . Holom orphic functional calculus Let 퐻 be a Hilbert space. In this section, we assume that 푇∶ 퐻 → 퐻 is a bounded linear operator. ℂ ∞ 푛 Let 푓 be a holomorphic function defined on such that 푓(푧) = ∑푛= 푎푛푧 , |푧| < 푅, where 푅 is the radius ‖푇푧‖퐻 of convergence of 푓. Let us also suppose 푅 > ‖푇‖, where ‖푇‖ = sup푧∈퐻⧵{ } ‖푧‖ is the operator norm of 푇. 0 퐻 푓(푇) A natural way of defining is the following one. It is clear that the series0 defined by ∞ 푛 푓(푇) = ∑ 푎푛푇 푛= is convergent in 퐵(퐻) and 푓(푇) ∈ 퐵(퐻). Moreover, if 푃‖푇‖0 denotes the space of holomorphic functions on {푧 ∈ ℂ ∶ |푧| < 푅} for some 푅 > ‖푇‖, the mapping 훹∶ 푃||푇|| ⟶ 퐵(퐻) 푓 ⟼ 푓(푇) is an algebra homomorphism. Notice that this way, the definition of 푓(푇) coincides with the natural definition whenever 푓 is a polynomial. There is also a natural way of defining 푓(푇) when 푓 is a rational function. Let 푝 푞 be polynomials such that 푞 푛 does not vanish in 휎(푇). Then, 푟 = 푝/푞 is a rational function with no poles in 휎(푇). Write 푞(푧) = ∏푗= (훼푗 −푧). − Since 훼푗 ∉ 휎(푇), we have that the resolvent operator 푅(훼푗 푇) = (훼푗퐼 − 푇) ,is a bounded operator on 퐻. Therefore, one can define 푟(푇) as 1 푛 1 , 푟(푇) = 푝(푇) ∏ 푅(훼푗 푇). 푗= Moreover, 푟(푇) ∈ 퐵(퐻) and the mapping 1 , 훹∶ 푅휍(푇) ⟶ 퐵(퐻) 푟 ⟼ 푟(푇) is and algebra homomorphism. Here 푅휍(푇) denotes the spaces of rational functions without poles in 휎(푇). Our objective is to define a functional calculus extending the two above particular cases (power series with a radius of convergence greater than ||푇|| and rational functions without poles in 휎(푇)). In order to do this, we consider holomorphic functions in some neighbourhood of the spectrum of 푇 ∈ 퐵(퐻). Let 푇 be a bounded operator in 퐻 and 훺 ⊂ ℂ an open set containing 휎(푇). For each connected component 훺푖 of 훺, we consider a closed contour 훾푖 in 훺푖 around 휎(푇) ∩ 훺푖. We let 훾 be the union of all the 훾푖. In particular, 훾 is a finite collection of smooth closed paths that is contained in 훺 ⧵ 휎(푇), see figure 1. Then, we have d푧 훼 ∈ 휎(푇) ∫ = { πi 훾 푧 − 훼 훼 ∉ 훺. 1 1, , Suppose that 푓 is a holomorphic function2 in 훺; in short,0, 푓 ∈ ℋ(훺).