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IX Escuela-Taller de Análisis Funcional (2019)

Holom orphic functional 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 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 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 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 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 defined on a subset of ℂ and 푇 is an operator defined on a Hilbert . 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 , ℱ denotes the 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.

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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 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 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). Motivated by the Cauchy formula, we define 푓(푇) = ∫푓(푧)푅(푧 푇) d푧 πi 훾 1 , , 2 TEMat monogr., 1 (2020) e-ISSN: 2660-6003 63 Holomorphic functional calculus

휎(푇)

F i g u r e 1 : The contour 훾 around 휎(푇). where 푅(푧 푇) = (푧퐼 − 푇)− is the resolvent operator. The integral is understood in the 퐵(퐻)-Böchner sense [3]. Since 푓 ∈ 퐻(훺) and1 훾 is contained in 훺 ⧵ 휎(푇), the Cauchy integral theorem and Hahn-Banach theorem imply, that the integral defining 푓(푇) does not depend on the contour 훾 satisfying the above conditions. Moreover, 푓(푇) ∈ 퐵(퐻).

T h e o r e m 1 . Let 푇 ∈ 퐵(퐻). Assume that 훺 is an open set in ℂ containing 휎(푇). Then, the mapping

훷∶ ℋ(훺) ⟶ 퐵(퐻) 푓 ⟼ 푓(푇) satisfies the following properties:

( 1 ) 훷 is an algebra homomorphism.

( 2 ) 훷(푝) = 푝(푇), for every polynomial 푝. ∞ ( 3 ) If {푓푛}푛= ⊂ ℋ(훺), 푓 ∈ ℋ(훺) and 푓푛 → 푓, as 푛 → ∞, uniformly on every compact subset of 훺, then 훷(푓푛) → 훷(푓) as 푛 → ∞ in 퐵(퐻). 1 Moreover, 훷 is the unique mapping from 퐻(훺) into 퐵(퐻) satisfying the properties (1), (2) and (3). This holomorphic functional calculus is also called the Dunford–Riesz functional calculus. Note thatthetwo ∞ 푛 previous examples are special cases of the Dunford–Riesz functional calculus. Indeed, if 푓(푧) = ∑푛= 푎푛푧 , ∞ 푛 with radius of convergence greater than ‖푇‖, by taking 훺 = 퐵( 푅), we have that 훷(푓) = ∑푛= 푎푛푇 . 0 Also, if 푟 ∈ 푅휍(푇) ∩ ℋ(훺) with 휎(푇) ⊂ 훺, and 푟 is a rational function without poles in 휎(푇), then 푛 − 푝(푧) 0 ℂ 훷(푟) = 푝(푇) ∏푖= (훼푖퐼 − 푇) . Here, we are assuming 푟(푧) = 푛 0, , where 푝(푧) is a polynomial over . ∏푖= (훼푖−푧) 1 The spectral 1mapping theorem holds for 훷, that is, if 훺 is an open1 set in ℂ containing 휎(푇) and 푓 ∈ 퐻(훺), then 푓(휎(푇)) = 휎(푓(푇)) [20, Theorem 10.28(b)]. We are going to give an idea on how to extend the holomorphic functional calculus to closed operators. Let 푇 be a closed operator and 훺 an open set such that 휎(푇) ⊂ 훺 and the resolvent set 휌(푇) ∩ (ℂ ⧵ 훺) ≠ ∅. We choose 훼 ∈ 휌(푇) ∩ (ℂ ⧵ 훺) and we consider the function

if 푧 ∈ ℂ ⧵ {훼} ⎧ 푧−훼 푅 (푧) = ∞1 if 푧 = 훼 훼 ⎨ ⎩ if 푧 = ∞. , , − It is clear that 푅훼(훺) = 푊 is an open set in ℂ ⧵0 { }. Also, we have that −푅훼(휎(푇)) = 휎((훼퐼 − 푇) ). Since − 휎(푇) ⊂ 훺, 휎((훼퐼 − 푇) ) ⊆ −푊. We define 1

1 0 − − 푓(푇) ≔ (푓 ∘ (−푅훼 )((훼퐼 − 푇) ) 1 1 where we use the holomorphic functional calculus defined in theorem 1 on the right hand side. ,

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3 . Sectorial operators

In this section we introduce sectorial operators and we discuss their main properties. These operators may not be defined on the whole Hilbert space 퐻 but on a dense subspace that we call domain of 푇 (and denote by 퐷(푇)). Moreover, they may be unbounded. For a (possibly unbounded) linear operator 푇∶ 퐷(푇) → 퐻, we define its spectrum 휎(푇) as the set of 휆 ∈ ℂ such that 푇 − 휆Id has a bounded inverse.

If < 휔 ≤ π, by 푆휔 we denote the open sector, that is symmetric with respect the positive real axis and with opening angle 휔, that is, 0 푆휔 = {푧 ∈ ℂ ⧵ { } ∶ |Arg(푧)| < 휔}.

Also, we define 푆 = ( ∞). 푆휔 denotes the closure of 푆휔, for every ≤ 휔 ≤ π. 0 D efinition 2 . Let0 푇∶ 퐷(푇) → 퐻 be a linear operator. We say that 푇 is a sectorial operator of angle 흎 ∈ [ π) (in short, 푇0, ∈ Sect(휔)) when the following two properties0 hold:

1 ) 휎(푇) ⊂ 푆 , 0, 푤 2 ) for every 휔 < 훼 < π, 푀(푇 훼) ≔ sup{‖푧 푅(푧 푇)‖ ∶ 푧 ∉ 푆훼} < ∞. ◀

Note that if 푇 ∈ Sect(휔) for some 휔 ∈ [ π), then (−∞ ) ⊆ 휌(푇) and 푇 is a closed operator. We name , , sectoriality angle 휔푇 of the operator 푇 to the number 0, , 0 휔푇 = min{ ≤ 휔 < π ∶ 푇 ∈ Sect(휔)}.

We now give a few examples of sectorial operators. 0

E x a m p l e 3 . If 푇 ∈ 퐵(퐻) is self-adjoint and positive, then 푇 ∈ Sect( ). ◀

π E x a m p l e 4 . Let < 휔 < . A family {푇(푧) ∶ 푧 ∈ 푆 } ⊆ 퐵(퐻) is said to be a holomorphic semigroup when 휔 0

1 ) 푇(푧 )푇(푧 ) = 푇(푧 + 푧 ), 푧 푧 ∈ 푆 , 0 2 푤 2 ) the mapping 1 2 1 2 1 2 , 훹∶ 푆휔 ⟶ 퐵(퐻) 푧 ⟼ 푇(푧)

is holomorphic. The infinitesimal generator of {푇(푧)} is the operator 퐴 defined by

푇(푡) − 퐼 퐴푥 = lim 푥 푥 ∈ 퐷(퐴) 푡→ + 푡

0 푇(푡)푥−푥 where the domain 퐷(퐴) of 퐴 is formed by all those 푥 ∈ 퐻, such that the, limit lim푡→ + 푡 exists. Then, π the operator −퐴 is in Sect( − 휔). ◀ 0

E x a m p l e 5 . We consider on2 ℂ the usual inner product defined by 2 (푢 푣) ⋅ (푎 푏) = 푢 ̄푎+ 푣푏̄ 푢 푣 푎 푏 ∈ ℂ.

We define the space ℓ (ℂ ) of 푧 = {푧 = (푢 푣 )}∞ in ℂ such that , , 푛 푛 , 푛 푛=, , , 2 2 2/ ∞ 1 ∞ , ‖푧‖ = ‖{푧푛}푛= ‖ ≔ (∑ (|푢푛| + |푣푛| )) < ∞. 1 2 푛= 2 2 2 1 2 The vector space ℓ (ℂ ) is actually a Hilbert space with1 the natural operations and the inner product of ℓ (ℂ ) is defined by 2 2 ∞ 2 2 ∞ ∞ 푧 ⋅ 푦 = {푧푛}푛= ⋅ {푦푛}푛= = ∑ 푧푛 ⋅ 푦푛̄ 푧 푦 ∈ ℓ (ℂ ). 푛= 2 2 1 1 , , TEMat monogr., 1 (2020) e-ISSN: 2660-60031 65 Holomorphic functional calculus

∞ We define the operator Ton ℓ (ℂ ) as follows: for 푧 = {(푢푛 푣푛)}푛= , we define ∞ 2 2 −푛 푢 푇(푧) = {( )( 푛 )}1 . −푛 , 푣 푛 푛= 2 1 Therefore, 푇 ∈ 퐵(ℓ (ℂ )). Note that 푇 is not a sectorial operator for any angle ≤ 휔 < 휋. Let 휀 < . Let 0 2 1 푧 = {(푢푛 푣푛)}푛 ∈ ℓ 2(ℂ 2). We have that ∞ 2 2 휀 − −푛 − 푢 0 0 (휀퐼 − 푇)(푧) = {( )( 푛 )} . , 휀 − −푛 푣 푛 푛= 2 1 And therefore, the resolvent operator has the following expression: 0 2 1 ∞ (휀 − −푛)− 푢 + (휀 − −푛)− 푣 (휀퐼 − 푇)− (푧) = {( 푛 푛 )} 푧 = {(푢 푣 )} ∈ ℓ (ℂ ) (휀 − −푛)− 푣 푛 푛 푛 1 푛 2 푛= 1 2 2 2 1 2 and , , , 2 1 ‖(휀퐼 − 푇)− ‖ ≥ sup (휀 − −푛)− = 휀− . 푛≥ 1 2 2 Hence, 푇 ∉ Sect(휔), for any 휔 ∈ [ π). ◀ 1 2

E x a m p l e 6 . The following differential operators are sectorial: 0, ″ 1 ) 푇푓 = −푓 , 푓 ∈ 퐷(푇) = 푊 (ℝ). ″ 2 ) 푇푓 = −푓 , 푓 ∈ 퐷(푇) = {푓2,2 ∈ 푊 ( ) ∶ 푓( ) = 푓( ) = }. ″ ′ ′ 3 ) 푇푓 = −푓 , 푓 ∈ 퐷(푇) = {푓 ∈ 푊 2,2( ) ∶ 푓 ( ) = 푓 ( ) = }. 0푛 , 1 0 1 0 4 ) Let 훺 be a bounded open set in2,2ℝ with 퐶 boundary and 0 , 1 0 1 0 푇푢 = −Δ푢 푢 ∈2 퐷(푇) = 푊 (훺) ∩ 푊 (훺).

2,2 1,2 0 Here, by 푊 we denote the usual Sobolev spaces, [20, Chapter 8.8]. ◀ Now, we present some of the main properties of sectorial operators.

Proposition 7 . Let 퐻 be a Hilbert space and 푇 ∈ Sect(휔), where ≤ 휔 ≤ π. The following statements hold.

− 1 ) If 푇 is injective, then 푇 ∈ Sect(휔). 0 1 푛 −푛 2 ) Let 푥 ∈ 퐻. We have that 푥 ∈ 퐷(푇) if and only if lim푡→∞ 푡 (푡 + 푇) 푥 = 푥. 푛 −푛 3 ) Let 푥 ∈ 퐻. Then, 푥 ∈ 푅(푇) if and only if lim푡→ 푇 (푡 + 푇) 푥 = 푥.

4 ) 퐷(푇) = 퐻 퐻 = 푁(푇) ⊕ 푅(푇) 푇 푅(푇) 퐻 and . is injective0 if and only if is dense in . ∗ 5 ) 푇 ∈ Sect(휔) and 휔푇 = 휔푇∗.

P r o o f .

− 1 ) Suppose that 푇 is injective. We have that 푇 ∶ 푅(푇) ⊂ 퐻 → 퐷(푇) ⊂ 퐻. If 휆 ≠ , 휆 ∈ 휎(푇) if and only − − − − − − if 휆 ∈ 휎(푇 ). Moreover 휆(휆 + 푇 ) = 퐼 − 1휆 ( 휆 + 푇) provided that 휆 ∈ 휌(푇). Then, 휎(푇 ) ⊂ 푆휔 − and,1 for every1 훼 ∈ (휔 π), 푀(푇 훼)1 <1 ∞. 1 1 1 1 0 1 −푛 1 푛 2 ) Since (푡 + 푇) 푥 ∈ 퐷(퐴), for every 푡 > , we deduce that 푥 ∈ 퐷(퐴) provided that 푥 = lim푡→+∞ 푡 (푡 + 푇)−푛푥. Suppose that ,푥 ∈ 퐷(푇). We, can write 0 푥 = 푡(푡 + 푇)− 푥 + (푡(푡 + 푇)− )푇푥. 푡 1 1 By iteration of this equality we get 1 푛 푥 = (푡(푡 + 푇)− )푛푥 + ∑ (푡(푡 + 푇)− )푘푇푥. 푡 푘= 1 1 − 1 − 푛 Since sup푡> ‖푡(푡 + 푇) ‖ < ∞, it follows that lim푡→+∞1 (푡(푡 + 푇) ) 푥 = 푥. By using again that − − 푛 1 1 sup푡> ‖푡(푡 + 푇) ‖ < ∞, we deduce that lim푡→+∞ (푡(푡 + 푇) ) 푦 = 푦, for every 푦 ∈ 퐷(푇). 0 1 1 66 0 https://temat.es/monograficos Ariza et al.

3 ) This property can be proved in a similar way as the previous one. − 4 ) ℕ ℕ Let 푥 ∈ 퐻. Since sup푛∈ℕ ‖푛(푛 + 푇) 푛‖ < ∞, there exists an increasing function 휙∶ → and − 푦 ∈ 퐻 such that 휙(푛)(휙(푛) + 푇) 푥 →1 푦, as 푛 → ∞, in the weak topology of 퐻. That is, 1 lim ⟨휙(푛)(휙(푛) + 푇)− 푥 푧⟩ = ⟨푦 푧⟩ 푧 ∈ 퐻. 푛→∞ 1 This implies that 푇(휙(푛) + 푇)− 푥 → 푥 − 푦, as 푛 → ∞, , in the, weak, topology of 퐻. Now, 푇 is a closed operator. This means that 퐺(푇)1 is a closed subspace of 퐻 × 퐻. Hence, 퐺(푇) is also weakly closed in 퐻 × 퐻. Then, 푥 = 푦. Since the closure of 퐷(푇) in the weak topology of 퐻 coincides with the closure of 퐷(푇) in H, we conclude that 푥 ∈ 퐷(푇). We now prove that 퐻 = 푁(푇) ⊕ 푅(푇). Note that, according to 2), if 푥 ∈ 푁(푇) ∩ 푅(푇), then

= 푇푥 = lim (푡 + 푇)− 푇푥 = lim 푇(푡 + 푇)− 푥 = 푥. 푡→ + 푡→ + 1 1 0 0 0 − Hence, 푁(푇) ∩ 푅(푇) = { }. Let 푥 ∈ 퐻. Since sup푡> ‖푡(푡 + 푇) 푥‖ < ∞, there exists a decreasing ∞ − sequence (푡푛)푛= ⊂ ( ∞) such that 푡푛 → , and 푡푛(푡푛 + 푇) 푥 →1 푦, as 푛 → ∞, in the weak topology 0 of 퐻. On the other hand,0 1 1 0, − 0 − 푡푛푇(푡푛 + 푇) 푥 = 푡푛(푥 − 푡푛(푡푛 + 푇) ) → as 푛 → ∞ 1 1 in 퐻. By recalling that 퐺(푇) is a weakly closed subspace of 퐻, it follows that 푇푦 = . Moreover, − 0, , 푇(푡푛 + 푇) 푥 → 푥 − 푦, in the weak topology of 퐻. Hence, 푥 − 푦 is in the weak closure of 푅(푇) that coincides with1 the closure of 푅(푇) in 퐻. We conclude that 푥 ∈ 푁(푇) + 푅(푇). 0 ∗ ∗ ∗ 5 ) This property follows by taking into account that 푝(푇 ) = {휆̄ ∶ 휆 ∈ 푝(푇)} and that 푅(휆 푇) = 푅(휆̄ 푇 ), for every 휆 ∈ 푝(푇). ▪ Before stating the last properties of sectorial operators, we give some definitions. , ,

D efinition 8 . A collection of operators {푇푖}푖∈퐼 is said to be uniformly sectorial of angle 흎 ∈ [ π) when ◀ 푇푖 ∈ Sect(휔), 푖 ∈ 퐼, and, for every 훼 ∈ (휔 π), sup푖∈퐼 푀(푇푖 훼) < ∞. ∞ 0, D efinition 9 . Suppose that the {푇 } is uniformly sectorial of angle 휔. We say that {푇 } , 푛 푛≤ , 푛 푛∈ℕ is a sectorial approximation on 푆휔 for an operator 푇 when there exists 휆 ∉ 푆휔 such that 휆 ∈ 휌(푇) and 1 푅(휆 푇푛) → 푅(휆 푇), as 푛 → ∞, in 퐵(퐻). In this case we write 푇푛 → 푇 (푆휔). ◀

Note that if {푇푛}푛∈ℕ is a sectorial aproximation on 푆휔 for 푇, then 푇 ∈ Sect(휔) and, for every 휆 ∉ 푆휔, , , , 푅(휆 푇푛) → 푅(휆 푇), in 퐵(퐻). In the next proposition we present some properties concerning sectorial convergence. , , ∞ Proposition 10 . Suppose that the sequence {푇푛}푛= uniformly sectorial of angle 휔.

− − 1 ) If 푇푛 → 푇 (푆휔) and 푇푛, 푇 are injective, then 1푇푛 → 푇 (푆휔).

2 ) If 푇푛 → 푇(푆휔) and 푇 ∈ 퐵(퐻), then there exists1 푛 ∈1ℕ such that 푇푛 ∈ 퐵(퐻) when 푛 ≥ 푛 and 푇푛 → 푇, as, 푛 → ∞ in 퐵(퐻). , ∞ 0 0 3 ) If {푇푛}푛= ⊂ 퐵(퐻), 푇 ∈ 퐵(퐻) and 푇푛 → 푇, as 푛 → ∞ in 퐵(퐻), then 푇푛 → 푇 (푆푤).

P r o o f . 1 ,

1 ) Follows from proposition 7.1). − − 2 ) Assume that 푇푛 → 푇 (푆휔), and 푇 ∈ 퐵(퐻). Then, −푅(− 푇푛) = (퐼 + 푇푛) → (퐼 + 푇) in 퐵(퐻). The set of invertible operators in 퐵(퐻) is open in 퐵(퐻). Also if 퐴푛, 푛 ∈ ℕ, and1 퐴 are in 퐵(퐻)1 , they are − − invertible and 퐴푛 →, 퐴, as 푛 → ∞ in 퐵(퐻), then 퐴푛 → 퐴1, as 푛 → ∞ in 퐵(퐻). Thus 2) is proved. ∞ 3 ) Assume that {푇푛}푛= ⊂ 퐵(퐻), 푇 ∈ 퐵(퐻) and 푇푛 →1 푇 as1푛 → ∞ in 퐵(퐻), 퐼 + 푇 is invertible and − − (퐼 + 푇푛) → (퐼 + 푇) , as 푛 → ∞ in 퐵(퐻). ▪ 1 1 1

TEMat monogr., 1 (2020) e-ISSN: 2660-6003 67 Holomorphic functional calculus

4 . Spaces of holom orphic functions

In this section we present the definitions and the main properties of the spaces of holomorphic functions which will be used to define holomorphic functional calculus in the next section. We recall that ℋ(훺) denotes the space of holomorphic functions on 훺.

Let 휑 ∈ ( π]. We say that a function 푓 ∈ ℋ(푆휑) is in the Dunford–Riesz class on 푆휑, shortly 푓 ∈ 풟ℛ(푆휑), when

0, ∞ i ) 푓 ∈ 퐻 (푆휑), that is, 푓 is bounded on 푆휑; 훼 i i ) there exists 훼 < such that 푓(푧) = 푂(|푧| ), as |푧| → ∞ with 푧 ∈ 푆휑; 훽 i i i ) there exists 훽 > such that 푓(푧) = 푂(|푧| ), as |푧| → with 푧 ∈ 푆휑. 0 If there is no possible confusion about the sector, we may write 풟ℛ instead of 풟ℛ(푆 ). Note that, if 0 0 휑 푓 ∈ 풟ℛ(푆휑), then the function 푔(푧) = 푓( /푧), 푧 ∈ 푆휑, is also in 풟ℛ(푆휑). In short, a holomorphic function is in 풟ℛ when it is bounded and tends to zero at the origin and at infinity sufficiently fast. Some examples of functions in 풟ℛ(푆휑) are the following:1 푧 π a ) 푓 (푧) = +푧 , 푧 ∈ 푆휑, < 휑 < . π −2푧 b ) 푓1 (푧) = 푧e , 푧 ∈ 푆휑, < 휑 ≤ . 1 0 2 −√푧 i휃 i휃/ c ) 푓2(푧) = √푧e , 푧 ∈ 푆휑, < 휑 ≤ π. Here, if 푧 = 푟e , with 푟 > and 휃 ∈ [−π π), then √푧 = √푟e . 0 2 2 In the following3 proposition, equivalent definitions of the functions in 풟푅(푆휑) are stated. The proof of these characterizations is straightforward0 from the definition. 0 ,

Proposition 11 . Let 휑 ∈ ( π]. Suppose that 푓 ∈ ℋ(푆휑). The following properties are equivalent:

1 ) 푓 ∈ 풟ℛ(푆 ). 휑 0, 푠 −푠 2 ) There exist 퐶 ≥ and 푠 > such that |푓(푧)| ≤ 퐶 min{|푧| |푧| }, 푧 ∈ 푆휑. |푧|푠 3 ) There exist 퐶 ≥ and 푠 > such that |푓(푧)| ≤ 퐶 , 푧 ∈ 푆 . 0 0 +|푧| 푠 , 휑 We now define another space of functions, namely 풟ℛ .2 A function 푓 ∈ ℋ(푆 ) is said to be in 풟ℛ (푆 ) 0 0 1 휑 휑 when 0 0 ∞ 1 ) 푓 ∈ 퐻 (푆휑) 훼 2 ) there exist 훼 < such that 푓(푧) = 푂(|푧| ), as |푧| → ∞ with 푧 ∈ 푆휑; ; 3 ) there exist 푟 > and 퐹 ∈ ℋ(퐵( 푟)) such that 퐹(푧) = 푓(푧), 푧 ∈ 푆휑 ∩ 퐵( 푟). 0 In other words, a function in 풟ℛ decays at infinity and is holomorphic in a neighborhood of the origin. 0 0, 0, Suppose that 푓 ∈ 풟ℛ(푆휑) + 풟ℛ 0(푆휑), that is 푓 = 푔 + ℎ with 푔 ∈ 풟ℛ(푆휑) and ℎ ∈ 풟ℛ (푆휑). Since ℎ can be extended as a holomorphic function to a neighborhood of the origin, there exists 푐 ∈ ℂ such that 0 0 ℎ(푧) = 푐 + 푂(|푧|), as 푧 → with 푧 ∈ 푆휑. On the other hand, if 푓 ∈ ℋ(푆휑) and 푐 ∈ ℂ, we can write

푐 푓(푧) − 푐 푧 0 푓(푧) = + + 푓(푧) 푧 ∈ 푆 . + 푧 + 푧 + 푧 휑

By taking these properties in mind, we can establish the following, proposition. 1 1 1 Proposition 12 . Let 휑 ∈ ( π]. Assume that 푓 ∈ ℋ(푆휑). The following assumptions are equivalent:

1 ) 푓 ∈ 풟ℛ(푆 ) + 풟ℛ (푆 ). 휑 0, 휑 ∞ 2 ) 푓 ∈ 퐻 and it satisfies that 0 훼 a ) there exists 훼 < such that 푓(푧) = 푂(|푧| ), as |푧| → ∞ with 푧 ∈ 푆휑; 훽 b ) there exist 훽 > and 푐 ∈ ℂ such that 푓(푧) = 푐 + 푂(|푧| ), as |푧| → with 푧 ∈ 푆휑. 0 0 0 68 https://temat.es/monograficos Ariza et al.

Note that the function 푓(푧) = is not in 풟ℛ(푆휑) + 풟ℛ (푆휑).

We introduce the last space of functions. By 풜(푆휑) we denote the function space formed by all those 0 −푛 푓 ∈ ℋ(푆휑) for which there exists1 푛 ∈ ℕ such that 푓(푧)( + 푧) ∈ 풟ℛ(푆휑) + 풟ℛ (푆휑). The functions in 풜(푆휑) can be characterized as shown in the next proposition. 0 1 Proposition 13 . Let 휑 ∈ ( π]. Suppose that 푓 ∈ ℋ(푆휑). The following properties are equivalent:

1 ) 푓 ∈ 풜(푆휑). 0, 2 ) 푓 has the following properties:

a ) 푓 ∈ ℋ푐(푆휑), that is, for every < 푟 < 푅 < ∞, 푓 is bounded on 푆휑 ∩ {푧 ∈ ℂ ∶ 푟 ≤ |푧| ≤ 푅}; 훼 b ) There exists 훼 < such that 푓(푧) = 푂(|푧| ), as 푧 → ∞ with 푧 ∈ 푆휑; 훽 c ) There exist 훽 > and 푐 ∈ ℂ such0 that 푓(푧) = 푐 + 푂(|푧| ), as 푧 → with 푧 ∈ 푆휑. 0 3 ) There exist 푐 ∈ ℂ, 푛 ∈ ℕ and 퐹 ∈ 풟ℛ(푆휑) such that 0 0 푛 푓(푧) = 푐 + ( + 푧) 퐹(푧) 푧 ∈ 푆휑.

If, in addition, 푓 is bounded, 3) holds with 푛 = . 1 , If 휔 ∈ [ π) and ℳ represents either 풟ℛ 풟ℛ , or 풜 we define 1 ℳ[푆 ] = ℳ(푆 ). 휔0 ⋃ 휑 0, , 휔<휑≤π

5 . Holom orphic functional calculus for sectorial operators

In this section we develop a holomorphic functional calculus for sectorial operators and functions in the spaces defined in the previous section. We will be able to give a meaning to the expression 푓(푇) when 푇 is a sectorial operator and 푓 is in the function spaces from the previous section. The basic tool is the Cauchy integral formula. We begin by defining the contours of the . + − Let 휑 ∈ ( π) and 훿 > . We define the path 훤휑 ≔ 훤휑 + 훤휑 , where + i휑 − −i휑 훤휑 (푡) = −푡e 푡 ∈ (−∞ ] and 훤휑 (푡) = 푡e 푡 ∈ ( ∞). 0, 0 Thus, 훤휑 is the boudnary of the sector 푆휑 oriented in the positive sense. We also consider the path + − , , 0 , , 0, 훤휑 훿 = 훤휑 훿 + 훤휑 훿 + 훤휑 훿, where

+ 0 i휑 i휃 − i휑 , 훤, 휑 훿(푡), = −푡e , 푡 ∈ (−∞ −훿] 훤휑 훿(휃) = 훿e 휃 ∈ (휑 π − 휑] 훤휑 훿(푡) = 푡e 푡 ∈ (훿 ∞). 0 See figure 2. Note that 훤휑 훿 is the boundary of 푆휑 ∪ 퐵( 훿) positively oriented. We can think that the paths , , , ; , , , 2 ; , , , 훤휑 and 훤휑 훿 go around the sector 푆휔 when < 휔 < 휑. , 0, , 훤휑 0 훤휑 훿

휑 , 휑 훿

F i g u r e 2 : The contours 훤휑 and 훤휑 훿, for 휑 = π/ .

, In the sequel of this section we assume that 푇 ∈ Sect(휔), where 휔 ∈ ( 3π].4 Our objective is to define the operator 푓(푇) for every 푓 ∈ 풜[푆휔]. Note that we cannot assure that 휎(푇) ⊂ 푆휑 when 휑 ∈ (휔 π].

We first define 푓(푇) when 푓 ∈ 풟ℛ(푆휑) and 휑 ∈ (휔 π]. 0, , TEMat monogr., 1 (2020) e-ISSN:, 2660-6003 69 Holomorphic functional calculus

Proposition 14 . Let 휑 ∈ (휔 π]. Suppose that 푓 ∈ 풟ℛ(푆휑). If 휔 < 휆 < 휑, the 퐵(퐻)-Bochner integral

, ∫ 푓(푧)푅(푧 푇) d푧 훤휆 converges absolutely (with the 퐵(퐻) norm). Moreover,, if 휔 < 휆 휆′ < 휑, then

∫ 푓(푧)푅(푧 푇) d푧 = ∫ 푓(푧)푅(,푧 푇) d푧. 훤휆 훤휆′

P r o o f . The convergence of the integral, can be shown using, proposition 11.2) and splitting the integral near zero and near infinity. The second statement follows from the Hahn-Banach theorem and theCauchy integral theorem. ▪

Following proposition 14, we define the following functional calculus. If 휑 ∈ (휔 π] and 푓 ∈ 풟ℛ(푆휑) we define 푓(푇) by 푓(푇) = ∫ 푓(푧)푅(푧 푇) d푧 , , πi 훤휆 1 where 휆 ∈ (휔 휑). Thus 푓(푇) ∈ 퐵(퐻) when 푓 ∈ 풟ℛ(푆휑) and 휑, ∈ (휔 ,π]. 2 Assume now that 푓 ∈ 풟ℛ (푆휑). We cannot ensure that the integral in proposition 14 converges if 푓( ) ≠ . We choose 훿, > so that 푓 can be holomorphically extended to 퐵(, 훿) ⧵ 푆휑. Proceeding similarly we can 0 define 0 0 0 푓(푇) = ∫ 푓(푧)푅(푧 푇) d푧0, πi 훤휆 훿 1 where 훿 휆 are as above, since one can check that the, integral is, absolutely, convergent. Thus, 푓(푇) ∈ 퐵(퐻). Note that if 푓( ) = , then 푓 ∈ 풟ℛ ∩ 풟ℛ and2 both definitions for 푓(푇) coincide, by the Cauchy integral theorem., If 푓 = 푔 + ℎ with 푔 ∈ 풟ℛ and ℎ ∈ 풟ℛ , we define 푓(푇) = 푔(푇) + ℎ(푇). This definition does 0 not depend on0 the0 choice of 푔 and ℎ. Some properties of the functional calculus we have defined are contained in the following proposition. 0

Proposition 15 . Let 휑 ∈ (휔 π].

1 ) Suppose that 푥 ∈ 푁(푇) and 푓 = 푔 + ℎ, where 푔 ∈ 풟ℛ(푆휑) and ℎ ∈ 풟ℛ (푆휑). Then, 푓(푇)(푥) = ℎ( )푥. , 2 ) The mapping 훹∶ 풟ℛ(푆휑) + 풟ℛ (푆휑) → 퐵(퐻) defined by 훹(푓) = 푓(푇) is a homomorphism of 0 algebras. 0 0 3 ) If 푓휇(푧) = 휇−푧 , where 휇 ∉ 푆휑, then 푓휇 ∈ 풟ℛ (푆휑) and 푓휇(푇) = 푅(휇 푇). 1 4 ) 퐴 ∈ 퐶(퐻) 푅(휆 푇) 휆 ∈ 휌(푇) 퐴푓(푇) = 푓(푇)퐴 Suppose that commutes with 0 for every . Then, for every 푓 ∈ 풟ℛ + 풟ℛ . , , P r o o f . 0

1 ) Since 푥 ∈ 푁(푇), we can write, for any 푧 ∈ 휌(푇), 푧푅(푧 푇)푥 = 푧푅(푧 푇)푥 − 푅(푧 푇)푇푥 = (푧푅(푧 푇) − 푇푅(푧 푇))푥 = (푧 − 푇)푅(푧 푇)푥 = 푥.

Then, by taking 휆 ∈ (휔 휑) and 훿 > such that ℎ can be holomorphically extended to 퐵( 훿) ⧵ 푆휑, , , , , , , 푓(푇)푥 = ∫ 푔(푧)푅(푧 푇)푥 d푧 + ∫ ℎ(푧)푅(푧 푇)푥 d푧 , πi 0 πi 0, 훤휆 훤휆 훿 1 1 푔(푧) , ℎ(푧), , = (2 ∫ d푧 + ∫2 d푧) 푥. πi 푧 πi 푧 훤휆 훤휆 훿 1 1 By using the Cauchy integral theorem, since 푔 ∈ 풟ℛ and, ℎ ∈ 풟ℛ , we conclude that 2 2 푔(푧) ℎ(푧) ∫ d푧 = and ∫ 0 d푧 = ℎ( ). πi 푧 πi 푧 훤휆 훤휆 훿 1 1 The first statement is proved. 0 , 0 2 2

70 https://temat.es/monograficos Ariza et al.

2 ) All the properties of an algebra homomorphism are straightforward to prove except for

(푓퐹)(푇) = 푓(푇) ∘ 퐹(푇)

for every 푓 퐹 ∈ 풟ℛ + 풟ℛ . The proof is easy but fairly long; it can be done by using some properties , of the resolvent operator and the Fubini and Cauchy integral theorems. 0 3 ) Let 휇 ∉ 푆휑,. It is clear that 푓휇 ∈ 풟ℛ (푆휑). We have that

0 푅(푧 푇) 푓 (푇) = ∫ d푧 휇 πi 휇 − 푧 훤휆 훿 1 , , , where 휆 ∈ (휔 휑) and 훿 < |휇|. The function 훹(2 푧) = 푅(푧 푇) is holomorphic in 휌(푇) taking values in 퐵(퐻). We finish by using the Hahn-Banach and Cauchy integral theorems.

4 ) Suppose that,푓 ∈ 풟ℛ(푆휑). In the general case we could, proceed in a similar way. We have that

푓(푇)푥 = ∫ 푓(푧)푅(푧 푇)푥 d푧 푥 ∈ 퐻 πi 훤휆 1 , , , where 휆 ∈ (휔 휑). Let 푥 ∈ 퐷(퐴). It follows2 that

, 푓(푇)퐴푥 = ∫ 푓(푧)푅(푧 푇)퐴푥 d푧. πi 훤휆 1 , The last 퐵(퐻)-Bochner integral is absolutely2 convergent in the 퐻−norm. The standard properties of the 퐵(퐻)-Bochner integral lead to 푓(푇)퐴푥 = 퐴푓(푇)푥. ▪

Finally, we define 푓(푇) for every 푓 ∈ 풜(푆휑). Let 푓 ∈ 풜(푆휑) with 휑 ∈ (휔 π]. There exists 푛 ∈ ℕ such that 푛 푓(푧)( + 푧) ∈ 풟ℛ + 풟ℛ . Therefore, we can define the operator 푓푛(푇) by 푓 , 0 푓 (푇) = (퐼 + 푇)푛 ( ) (푇) 1 푛 푝푛 where 푝(푧) = + 푧. We make two small remarks about this definition., Note firstly that, since (퐼 + 푇)−푛 ∈ 푛 푛 퐵(퐻), (퐼 + 푇) is a closed operator. Then, the operator 푓푛(푇) is closed because (푓/푝 )(푇) ∈ 퐵(퐻). Secondly, −푛 one can prove1 that the definition of 푓푛 does not really depend on 푛, as long as 푓푝 ∈ 풟ℛ + 풟ℛ . The proof is not difficult but it is a bit technical, so we omit it. Therefore, wedefine 푓(푇) = 푓푛(푇) for any admissible 푛. 0 Some properties of this functional calculus are summarized in the following proposition. For the sake of conciseness, we omit the proof, which is easy but tedious.

Proposition 16 . Let 휑 ∈ (휔 π], and 푓 ∈ 풜(푆휑).

i ) If 푇 ∈ 퐵(퐻), then 푓(푇) ∈ 퐵(퐻). , i i ) If 푆 ∈ 퐵(퐻) and 푆 commutes with 푇, then 푆 commutes with 푓(푇).

i i i ) Suppose that 푔 ∈ 풜(푆휑). We have that 푓(푇) + 푔(푇) ⊂ (푓 + 푔)(푇) and 푓(푇)푔(푇) ⊂ (푓푔)(푇).

Furthermore, 퐷((푓푔)(푇)) ∩ 퐷(푔(푇)) = 퐷(푓(푇)푔(푇)). We cannot say that the natural functional calculus commutes with the sum and the product of functions (see proposition 16.iii)). But if 푓 푔 ∈ 풜(푆휑), we can see that

푓(푇) + 푔(푇) = (푓 + 푔)(푇) , 푓(푇)푔(푇) = (푓푔)(푇) , 푛 provided that 푔(푇) ∈ 퐵(퐻). For instance, if 푓(푧) = 푐 + ( + 푧) 푔(푧), 푧 ∈ 푆휑, where 푐 ∈ ℂ, 푛 ∈ ℕ, and 푛 , 푔 ∈ 퐷푅(푆휑), then 푓(푇) = 푐 + ( + 푇) 푔(푇). 1 TEMat monogr., 1 (2020)1 e-ISSN: 2660-6003 71 Holomorphic functional calculus

∗ If < 휑 ≤ π and 푓 ∈ 퐻(푆휑), we define the function 푓 by

∗ 0 푓 (푧) = 푓(푧) 푧 ∈ 푆휑. Thus, 푓∗ ∈ ℋ(푆 ), where 푓∗ is named the conjugate of the function 푓. It is clear that the function spaces 휑 , that we have considered in section 4 are invariant with respect to conjugation. In the following proposition we show how the natural functional calculus acts on adjoints.

∗ ∗ ∗ Proposition 17 . Let 휑 ∈ (휔 π] and 푓 ∈ 풜(푆휑). Then, 푓(푇 ) = (푓 (푇)) .

P r o o f . Suppose firstly that 푓 ∈ 풟ℛ(푆 ). We have that , 휑 푓∗(푇) = ∫ 푓∗(푧)푅(푧 푇) d푧 ∈ 퐵(퐻). πi 훤휆 1 Here 휔 < 휆 < 휑. For every 푥 푦 ∈ 퐻, , 2

⟨푓∗(푇)푥 푦⟩ = ⟨ ∫ 푓(푧)푅(푧 푇)푥 d푧 푦⟩ , πi 훤휆 1 , = ∫ 푓(푧)⟨푅(푧,푇)푥 푦⟩,d푧 2πi 훤휆 1 = ∫ 푓(푧)⟨푥 푅(, 푧 푇,∗)푦⟩ d푧 2πi 훤휆 1 = ⟨푥 ∫ 푓(푧,)푅(푧, 푇∗)푦 d푧⟩ 2 πi 훤휆 1 ∗ = ⟨푥,푓(푇 )푦⟩. , 2 We have taken into account that all the Böchner integrals that appear are absolutely convergent. Then, , (푓∗(푇))∗ = 푓(푇∗).

If 푓 ∈ 풟ℛ (푆휑) we can proceed in a similar way. Assume now that 푓 ∈ 풜(푆휑). There exist 푛 ∈ ℕ, 푓(푧) 푔 ∈ 풟ℛ(푆휑) and ℎ ∈ 풟ℛ (푆휑) such that 푛 = 푔(푧) + ℎ(푧), 푧 ∈ 푆휑. Then, 0 ( +푧) ∗ ∗ 푛 ∗ 0 푓(푇1 ) = ( + 푇 ) 퐹(푇 ) where 퐹 = 푔 + ℎ. Here (퐹∗(푇))∗ = (푔∗(푇))∗ + (ℎ∗(푇))∗ ∈ 퐵(퐻). Moreover, we can write 1 , ( + 푇∗)푛 = (− )푛( + 푇∗)푛[푅(− 푇))푛]∗[( + 푇)푛]∗ = (− )푛( + 푇∗)푅(− 푇∗)푛[( + 푇)푛]∗ = [( + 푇)푛]∗. 1 1 1 1, 1 We get 1 1 1, 1 1 푓(푇∗) = [( + 푇)푛]∗(퐹∗(푇))∗ = (퐹∗(푇)( + 푇)푛)∗. According to proposition 15.4), for every 푥 ∈ 퐷(푇푛) = 퐷(( + 푇)푛), we have 1 1 퐹∗(푇)( + 푇)푛푥 = ( + 푇)푛퐹∗(푇)푥 1 or, in other words, 퐷(푇푛) ⊂ 퐷(푓∗(푇)). Since 퐷(푇) = ℋ, using proposition 7.2), we deduce that (푡 + (푡 + 1 1 , 푇)− )푛푥 → 푥, as 푡 → ∞, and 푓∗(푇)(푡+(푡+푇)− )푛푥 = (푡(푡+푇)− )푛푓∗(푇)푥 → 푓∗(푇)푥, as 푡 → ∞. Furthermore, − 푛 푛 (푡 +1 (푡 + 푇) ) 푥 ∈ 퐷(푇 ), for every 푡 > . 1 1 1 ∗ ∗ This fact allows us to conclude that the closure 푓 (푇)|퐷(푇푛) = 푓 (푇) can be written as 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 푓(푇 ) = (푓 (푇)|퐷(푇푛)) = (푓 (푇)|퐷(푇푛)) = (푓 (푇)) . ▪

We can find in the literature extensions of the natural functional calculus. There exists a trade-off between operators and function spaces. When the operator is better we can define a functional calculus for a wider class of functions and vice-versa. We finish this section with the following definition.

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∞ D efinition 18 . Let 휑 ∈ (휔 π]. Suppose that ℱ is an algebra contained in 풜(푆휑) ∩ 퐻 (푆휑). The natural functional calculus on ℱ for 푇 is said to be bounded when 푓(푇) ∈ 퐵(퐻), for every 푓 ∈ ℱ, and there exists 퐶 > such that , ‖푓(푇)‖ ≤ 퐶‖푓‖∞ ∀푓 ∈ ℱ. ◀ 0 ∞, 6 . An operatorwithout a bounded 퐻 - c a l c u l u s

McIntosh and Yagi [18] presented the first example of a sectorial operator without a bounded 퐻∞-calculus. We are going to give an example that can be found in the work of Le Merdy [11] (see also Haase’s thesis [7, p. 49]). Suppose that 퐻 is a separable Hilbert space. We can think for instance 퐻 = ℓ (ℕ). We say that a sequence ∞ {푒푛}푛= in 퐻 is a conditional basis in 퐻 when the following two properties hold:2 ∞ ∞ i ) ℂ 1For every 푥 ∈ 퐻 there exists a unique sequence (휇푛)푛= ⊂ such that 푥 = ∑푛= 휇푛푒푛. ∞ ∞ i i ) There exist a sequence (휇푛)푛= ⊂ ℂ of complex numbers and a sequence (휃푛)푛= ⊂ {− } of signs ∞ ∞ 1 1 such that ∑푛= 휇푛푒푛 converges in 퐻, but ∑푛= 휃푛휇푛푒푛 does not converge in 퐻. 1 1 1, 1 ∞ According to Lindenstrauss1 and Tzafriri [16, Proposition1 2.b.11], there exists a conditional basis {푒푛}푛= of 퐻. We may assume that ‖푒푛‖ = for all 푛 ∈ ℕ. We define the 푛-th projection operator 푃푛 in the standard way: 1 1 푃푛 ∶ 퐻 ⟶ 퐻 ∞ 푛

푥 = ∑ 휇푘푒푘 ⟼ 푃푛(푥) = ∑ 휇푘푒푘. 푘= 푘= ∞ For every 푛 ∈ ℕ, 푃푛 ∈ 퐵(퐻). Also, one can prove1 that the sequence1 {푃푛}푛= is bounded in 퐵(퐻). The constant ∞ 푀퐷 = sup푛∈ℕ ‖푃푛‖ is known as the constant basis for {푒푛}푛= . This constant plays an important role inthe 1 theory of basis, but we will only use that 푀퐷 is finite. 1 ∞ Let 푎 = (푎푛)푛= be a complex sequence. We define the multiplier operator associated to 푎 as follows: if ∞ ∞ 푥 = ∑푛= 휇푛푒푛 ∈ 퐻 is such that ∑푛= 푎푛휇푛푒푛 converges in 퐻, then we set 1 ∞ 1 1 푇푎푥 = ∑ 푎푛휇푛푒푛. 푛=

These multipliers operators were studied by Venni [231 ]. Note that the domain ∞ ∞

퐷(푇푎) = {∑ 휇푛푒푛 ∶ ∑ 푎푛휇푛푒푛 ∈ 퐻} 푛= 푛=

푀 1 1 ℂ ℕ is dense in 퐻 because the dense set { ∑푛= 휇푛푒푛 ∈ 퐻 ∶ 휇푛 ∈ 푀 ∈ } is contained in 퐷(푇푎).

Proposition 19 . The multiplier operator 푇 is a closed operator. 푎1 , ∞ P r o o f . Suppose that {푥푘}푘= ⊂ 퐷(푇푎) is such that 푥푘 → 푦 and 푇푎푥푘 → 푧 as 푘 → ∞, for some 푦 푧 ∈ 퐻. Then,

푘→∞ 1 푃 (푥푘) −−−→푃 푦 , 푘→∞ 푃 (푇 푥 ) = 푎1푃 푥 −−−→푃1푧 푎 푘 푘 , and, in general, for every 푛 ∈ ℕ 1 1 1 1 , 푘→∞ 푃푛+ (푥푘) − 푃푛(푥푘) −−−→푃푛+ (푦) − 푃푛(푦) 푘→∞ 푃 (푇 푥1 ) − 푃 (푇 푥 ) −−−→푃 1(푧) − 푃 (푧). 푛+ 푎 푘 푛 푎 푘 푛+ 푛 ,

We deduce that 푎 푃 (푦) = 푃 (푧) and,1 for every 푛 ∈ ℕ, 푎푛+ (푃푛+ (푦)−푃1 푛(푦)) = 푃푛+ (푧)−푃푛(푧). Hence 푦 ∈ 퐷(푇푎) and 푇푎푦 = 푧. ▪ 1 1 1 1 1 1

TEMat monogr., 1 (2020) e-ISSN: 2660-6003 73 Holomorphic functional calculus

∞ Let (푎푛)푛= be a sequence of complex numbers. We define

∞ 1 ‖푎‖ = lim sup |푎푛| + ∑ |푎푛+ − 푎푛|. 푛→∞ 푛= 1 ∞ 1 ∞ This quantity can be infinite. When ∑푛= |푎푛+ − 푎푛| < ∞ we say that (푎푛)푛= has bounded variation, and ∞ in this case (푎푛)푛= is convergent. 1 1 1 Proposition 20 . Let1 푎 = (푎푘) be a complex sequence and 푇푎 the associated multiplier operator. If ‖푎‖ < ∞, then 푇푎 ∈ 퐵(퐻) and ‖푇푎‖ ≤ 푀 ‖푎‖.

∞ ∞ P r o o f ℂ ℕ . Suppose that 푥 = ∑푘=0휇푘푒푘 ∈ 퐻 with (휇푘)푘= ⊂ . For 푛 ∈ , we have

푛 1 푛 1 푛− ∑ 푎푘휇푘푒푘 = ∑ 푎푘(푃푘 − 푃푘− )푥 + 푎 푃 푥 = ∑ (푎푘 − 푎푘+ )푃푘푥 + 푎푛푃푛푥. 1 푘= 푘= 푘= 1 1 1 1 ∞ 1 ∞ 1 ∞1 Since ‖푎‖ < ∞, {푎푛푃푛푥}푛= converges and {∑푘= (푎푘 −푎푘+ )푃푘푥}푘= is absolutely convergent in norm. Hence, ∞ ∑푘= 푎푘휇푘푒푘 converges and 1 ‖ ∞ 1 ‖ 1 1 1 ‖∑ 푎푘휇푘푒푘‖ ≤ 푀 ‖푎‖‖푥‖. ▪ ‖푘= ‖ 0 Let 휆 ∈ ℂ such that 휆 ≠ 푎푛, 푛 ∈ ℕ. Then,1 the multiplier operator associated to the sequence 휆 − 푎 = ∞ − {휆 − 푎푛}푛= is 푇휆−푎 = 휆퐼 − 푇푎 and it is injective. Moreover, (휆퐼 − 푇푎) is the multiplier operator associeated ∞ to the sequence { }푛= . 휆−푎푛 1 1 1 ∞ Proposition 21 . Let 푎 = (푎1 푛)푛= be an increasing sequence of real numbers satisfying that 푎 > and lim푛→∞ 푎푛 = ∞. Then, the associated multiplier operator 푇푎 is sectorial of angle . 1 1 ∞ 0 P r o o f . Let 휆 ∈ ℂ ⧵ [푎 ∞). We define 푎(휆) = ( )푛= . We can write 휆−푎푛 0 1 ∞ ∞ 푎 ∞ 1 | | 1 | 푛+ d푡 | d푡 ‖푎(휆)‖, = ∑ | − | = ∑ |∫ | ≤ ∫ . | 1 | |휆 − 푎푛+ 휆 − 푎푛 | (휆 − 푡) |휆 − 푡| 푛= 푛= | 푎푛 | 푎 1 1 2 2 1 1 1 i휃 1 Hence, 휆 ∈ 휌(푇푎). We conclude that 휎(푇푎) ⊂ [푎 ∞). Also, if 휆 = 푟e , with 푟 > and 휈 ∈ [−π π) we get

∞ ∞ 1 푟 d푡 d푢 ‖휆(휆퐼 − 푇 )− ‖ ≤ 푀 |휆|‖푎(휆)‖, ≤ 푀 ∫ = 푀 ∫ 0 . , 푎 푟ei휃 − 푡| |ei휃 − 푢| 1 0 0 0 π 2 2 If 휑 ∈ ( ) we have that 0 0 ∞ d푢 ∞ d푢 ∫ = ∫ 0, 2 |ei휃 − 푢| (cos 휃 − 푢) + sin 휃 is bounded by 2 2 2 0 0 ∞ d푢 ∫ for 휃 ∈ [π/ π − 휑] ∪ [휑 − π −π/ ] 푢 + sin 휑 ∞ 2 d푢 2 ∫0 , for 휃 ∈ (π −2, 휑 π) ∪ [−π 휑 −, π] 2 ; (cos 휑 + 푢) cos 휃 d푢 2 ∞ d푢 ∫0 + ∫, , , ; 2 sin 휑 cos 휃 (cos 휃 − 푢) + sin 휑 ∞ cos2 휃 d푢 2 2 0 ≤ + ∫2 for 휃 ∈ [휑 π/ ] ∪ [−π/ −휑]. sin 휑 푢 + sin 휑

2 2 1 2 2 − 0 4 , , 2 2, Hence, ‖휆(휆퐼 − 푇푎) ‖ ≤ 퐶휑, 휆 ∉ 푆휑. Thus, we prove that 푇푎 ∈ Sect( ). ▪ 1 0 74 https://temat.es/monograficos Ariza et al.

ξλ,δ

δ λ • a1

F i g u r e 3 : The contour 휉휆 훿 and 푎 .

, 1 푛 ∞ Finally, let us show that if 푎 = ( )푛= , then the operator 푇푎 does not have a bounded functional calculus. ∞ −푚 Let 푓 ∈ 풜(푆휑) ∩ 퐻 (푆휑) where 휑 ∈ ( π]. There exists 푚 ∈ ℕ such that 푓(푧)( + 푧) = 푔(푧) ∈ 풟ℛ + 풟ℛ . 1 Since 휎(푇푎) ⊂ [푎 +∞) we can write2 0 0, 1 1 푔(푇 ) = ∫ 푔(푧)푅(푧 푇 ) d푧 , 푎 πi 푎 휉휆 훿 1 + − , , , where the path 휉휆 훿 = 휉휆 훿 + 휉휆 훿 + 휉휆 훿, with 2< 휆 < 휑, < 훿 < 푎 and

+ i휆 0 − −i휆 −i휃 휉 (푡) =, −푡e , 푡 ∈ (−∞, −훿], 휉 (푡) = 푡e 푡 ∈ [훿 ∞)1 휉 (푡) = 훿e 휃 ∈ (−휆 +휆). 휆 훿 휆 훿 0 0 휆 훿 See figure 3. By using the Hahn-Banach and Cauchy integral theorems0 we deduce that , , , , , , , , , , , 푔(푧) 푔(푇푎)(푒푛) = ∫ 푔(푧)푅(푧 푇푎)푒푛 d푧 = ∫ d푧 푒푛 = 푔(푎푛) 푒푛. πi πi 푎푛 − 푧 휉휆 훿 휉휆 훿 1 1 푛 ℂ ℕ , , , Then, if (휇푘)푘= ⊂ with 푛 ∈2 , 2 푛 푛 푛 1 푚 푓(푇푎) (∑ 휇푘푒푘) = ∑ 휇푘( + 푇) 푔(푎푘) 푒푘 = ∑ 휇푘푓(푎푘) 푒푘. 푘= 푘= 푘= ∞ If the natural functional calculus on1 풜(푆휑)∩퐻1 (푆휑1) for 푇푎 were bounded,1 then we would have 푓(푇푎) ∈ 퐵(퐻) ∞ ∞ and, hence, the series ∑푘= 휇푘푓(푎푘)푒푘 would converge in 퐻 provided that ∑푘= 휇푘푒푘 ∈ 퐻. 푛 Now, take the sequence 푎푛 = , 푛 ∈ ℕ. Since {푒푛} is a conditional basis, there exist two sequences ∞ ℂ ∞ 1 ∞ ∞ 1 (휇푛)푛= ⊂ and (휃푛)푛= ⊂ {− } such that ∑푛= 휇푛푒푛 converges in 퐻 and ∑푛= 휃푛휇푛푒푛 does not converge ∞ 푛 in 퐻. According to Garnett [6, Theorem2 1.1, VII.1], there exists 푓 ∈ 퐻 (푆휋/ ) ∩ 풜(푆휋/ ) such that 푓( ) = 휃푛, 1 1 ∞ 푛 ∈ ℕ. We conclude that, if 푇푎1,is 1 the multiplier operator1 associated with {푎푛}푛= 1, 푓(푇푎) ∉ 퐵(퐻). Hence, the ∞ 2 2 natural functional calculus on 풜(푆휑) ∩ 퐻 (푆휑) for 푇푎 is not bounded. 2 1 Multiplier operators like those we have just studied have also been considered to obtain examples related to certain operator theoretical problems [2, 9, 10, 12, 21, 23].

R e f e r e n c e s

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[3] BOCHNER, Salomon. “Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind”. In: Fundamenta Mathematicae 20.1 (1933), pp. 262–176. URL: http://eudml.org/doc/212635. [4] COWLING, Michael; DOUST, Ian; MCINTOSH, Alan, and YAGI, Atsushi. “ operators with a bounded 퐻∞ functional calculus”. In: Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 60.1 (1996), pp. 51–89. ISSN: 0263-6115. https://doi.org/10.1017/ S1446788700037393. [5] DUNFORD, Nelson and SCHWARTZ, Jacob T. Linear operators. Part 1. Wiley Classics Library. New York: John Wiley & Sons, Inc., 1988. ISBN: 978-0-471-60848-6. [6] GARNETT, John B. Bounded analytic functions. 1st ed. Graduate Texts in Mathematics 236. New York: Springer, 2007. https://doi.org/10.1007/0-387-49763-3. [7] HAASE, Markus. The functional calculus for sectorial operators and similarity methods. PhD thesis. Fakultät für Mathematik, Universität Ulm, 2003. [8] HAASE, Markus. “A general framework for holomorphic functional calculi”. In: Proceedings of the Edinburgh Mathematical Society 48.2 (2005), pp. 423–444. ISSN: 0013-0915. https://doi.org/10. 1017/S0013091504000513. [9] KALTON, Nigel. “Applications of Banach space theory to sectorial operators”. In: Recent Progress in Functional Analysis. International Functional Analysis Meeting on the Occasion of the 70th Birthday of Professor Manuel Valdivia (Valencia, 2000). North-Holland Mathematics Studies 189. Elsevier, 2001, pp. 61–74. https://doi.org/10.1016/s0304-0208(01)80036-0. [10] LANCIEN, Gilles. “Counterexamples concerning sectorial operators”. In: Archiv der Mathematik 71.5 (1998), pp. 388–398. ISSN: 0003-889X. https://doi.org/10.1007/s000130050282. [11] LE MERDY, Christian. “퐻∞-functional calculus and applications to maximal regularity”. In: Semi- groupes d’opérateurs et calcul fonctionnel (Besançon, 1998). Publications mathématiques de l’UFR Sciences et techniques de Besançon 16. Besançon: Université de Franche-Comté, 1999, pp. 41–77. [12] LE MERDY, Christian. “A bounded compact semigroup on Hilbert space not similar to a contraction one”. In: Semigroups of operators: theory and applications (Newport Beach, 1998). Progress in Nonlinear Differential Equations and Their Applications 42. Basel: Birkhäuser, 2000, pp. 213–216. [13] LE MERDY, Christian. “The Weiss conjecture for bounded analytic semigroups”. In: Journal of the London Mathematical Society. Second Series 67.3 (2003), pp. 715–738. ISSN: 0024-6107. https://doi. org/10.1112/S002461070200399X. [14] LE MERDY, Christian. “On square functions associated to sectorial operators”. In: Bulletin de la Société Mathématique de France 132.1 (2004), pp. 137–156. ISSN: 0037-9484. https://doi.org/10.24033/ bsmf.2462. [15] LE MERDY, Christian. “Square functions, bounded analytic semigroups, and applications”. In: Per- spectives in . Banach Center Publications 75. Warsaw: Institute of Mathematics, Polish Academy of Sciences, 2007, pp. 191–220. https://doi.org/10.4064/bc75-0-12. [16] LINDENSTRAUSS, Joram and TZAFRIRI, Lior. Classical Banach spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 92. Berlin, New York: Springer, 1977. ISBN: 978-3-540-08072-5.

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