On the spectra of algebras of analytic functions

Carando, Daniel; Garcı́a, Domingo; Maestre, Manuel; Sevilla‐Peris, Pablo

doi:10.1090/conm/561/11114

Cited by 12 publications

(9 citation statements)

References 22 publications

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“…The purpose of this note is to exhibit that the Cluster Value Theorem holds for every Banach space X whose dual has the bounded approximation property for the algebras A u (B X ), H b (X) and to several open problems in the area (in particular, it gives answers to the questions posed in [17,Remark 2.5] and also [11,Questions 7 and 8]).…”

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confidence: 99%

Cluster values for algebras of analytic functions

Carando

Galicer

Muro

et al. 2018

Advances in Mathematics

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The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX ), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX ; and also for the Fréchet algebra H b (X) of holomorphic functions of bounded type on X (more generally, for H b (U ), the algebra of holomorphic functions of bounded type on a given balanced open subset U ⊂ X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space.As a consequence , we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras.2010 Mathematics Subject Classification. Primary 46J15, 46E50, 30H05.

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confidence: 99%

Cluster values for algebras of analytic functions

Carando

Galicer

Muro

et al. 2018

Advances in Mathematics

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“…The first steps towards the description of the spectrum of H b (X) were taken by Aron, Cole and Gamelin in [6]. A survey with the most relevant recent developments on the research of the spectra of algebras of analytic functions can be found in [54]. We also show, as in the case of H b (X) and HW (X), that any function f ∈ V H(X) extends naturally to an analytic function defined on the spectrum.…”

Section: Introductionmentioning

confidence: 99%

Operators on wighted spaces of holomorphic functions

Meneu¹,

María²

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que la presente memoria "Operators on weighted spaces of holomorphic functions" ha sido realizada bajo nuestra dirección por María José Beltrán Meneu y constituye su tesis para optar al grado de Doctor en Ciencias Matemáticas, con mención "Doctor Internacional".Y para que así conste en cumplimiento de la legislación vigente presentamos y apadrinamos ante la Escuela de Doctorado de la Universitat Politècnica de València la referida tesis firmando el presente certificado. Valencia, Febrero de 2014Los directores:José Bonet Solves y Carmen Fernández Rosell a la memòria de mon tio Juan Alfonso a la meua família v Agraïments M'agradaria dedicar unes paraules a totes aquelles persones que, amb el seu recolçament incondicional, m'han acompanyat durant aquest capítol de la meua vida. Sense elles, aquesta tesi no hagués estat possible.Fent balanç d'aquests anys puc dir amb total convenciment que l'entorn matemàtic que m'ha envoltat és molt difícil de superar. En primer lloc, vull expressar la meua gratitut als meus directors de tesi, els professors José Bonet i Carmina Fernández, per dipositar en mi la seua confiança i per tot el temps que m'han dedicat. D'una banda, la seua accessibilitat, professionalitat i amor per les matemàtiques, i d'una altra, la seua proximitat i comprensió, sobretot en els moments personals més difícils, han sigut fonamentals perquè aquest projecte haja pogut dur-se a terme. Haver pogut treballar amb professionals del seu nivell ha sigut una gran sort. I would also like to thank professor Angela Albanese for being my advisor in my stay in Lecce. She, together with professor Elisabetta Mangino and the PhD students from "Università del Salento" were able to create a very nice working environment in which I felt very welcome.Gràcies també a tots els professors de matemàtiques que he tingut al llarg de la meua vida, especialment a José Luis Morera, professor de l'Escola Tècnica Superior d'Arquitectura, qui va transmetre'm un entusiasme per aquesta disciplina suficient com per decidir canviar de carrera. També als professors Antonio Galbis, Carmina Fernández i Pablo Galindo, decisius a l'hora de decantar-me per l'anàlisi funcional i la variable complexa.La tesi és un procés llarg i costós, de vertiginosa especialització, i no són pocs els moments en què et replanteges si el temps dedicat val la pena. Per això, comptar amb un marc favorable al teu voltant, tant de "coaches" que t'aconsellen i motiven, com d'iguals amb qui compartir dubtes i misèries, és molt important. I jo l'he tingut. Recorde amb enyorança entrar cada matí al IUMPA i anar recorrent tots els despatxos per tal de saludar i parlar un poquet amb tots els companys. I és que es podria dir que la gent del IUMPA forma una gran família. A tots vii els germans grans, agrair-los la seua ajuda i bons consells, tan a nivell matemàtic com personal, especialment a David, per la seua disponibilitat a l'hora de resoldre dubtes de L A T E X, a Enrique, pels seus "on fire" matemàtics registrats en infinitat de e-mails, els quals m'han aportat un gran n...

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“…We refer to [8,11] for general theory on complex analysis in Banach spaces, and to [5,6,8,10,13] for background on the space of holomorphic functions of bounded type and its spectrum.…”

Section: Introductionmentioning

confidence: 99%

The algebra of bounded-type holomorphic functions on the ball

Carando

Muro

Vieira

2020

Proc. Amer. Math. Soc.

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We study the spectrum M b (U ) of the algebra of bounded type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space E as an analytic manifold over the bidual of the space. In the case that U is the unit ball of ℓ p , 1 < p < ∞, we prove that each connected component of M b (B ℓp ) naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are strictly smaller than one, and can be arbitrarily small. We also show that for other Banach sequence spaces, connected components do not necessarily identify with balls.2000 Mathematics Subject Classification. Primary 46G20, 46E50, 46T25, 46E25. Secondary 58B12, 32D26, 32A38.

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On the spectra of algebras of analytic functions

Cited by 12 publications

References 22 publications

Cluster values for algebras of analytic functions

Cluster values for algebras of analytic functions

Operators on wighted spaces of holomorphic functions

The algebra of bounded-type holomorphic functions on the ball

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