Uniform Algebras Generated by Holomorphic and Pluriharmonic Functions

Izzo, Alexander J.

doi:10.2307/2154301

Cited by 7 publications

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“…In the one-variable case, Theorem 1.1 is due to Čirca [3] (see also Axler-Shields [1]). Theorems 1.1 and 1.3 generalize work by A. Izzo [7] and Weinstock [13]. Theorem 1.2 generalizes Wermer's maximality theorem to C 2 to the effect that analyticity is the only obstruction to the full algebra being generated.…”

Section: Introductionmentioning

confidence: 71%

Uniform algebras and approximation on manifolds

Samuelsson

Wold

2011

Invent. math.

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Abstract.Let Ω ⊂ C n be a bounded domain and let A ⊂ C(Ω) be a uniform algebra generated by a set F of holomorphic and pluriharmonic functions. Under natural assumptions on Ω and F we show that the only obstruction to A = C(Ω) is that there is a holomorphic disk D ⊂ Ω such that all functions in F are holomorphic on D, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.

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Section: Introductionmentioning

confidence: 71%

Uniform algebras and approximation on manifolds

Samuelsson

Wold

2011

Invent. math.

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“…There is a large literature of related approximation theorems; see, for example, Čirka [5], Axler and Shields [2], Izzo [9].…”

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confidence: 99%

Commutants of analytic Toeplitz operators on the Bergman space

Axler

Čučković²,

Rao³

1999

Proc. Amer. Math. Soc.

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Abstract. In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.Let Ω be a bounded open domain in the complex plane and let dA denote area measure on Ω. The Bergman space L 2 a (Ω) is the subspace of L 2 (Ω, dA) consisting of the square-integrable functions that are analytic on Ω. For a bounded measurable function ϕ on Ω, the Toeplitz operator T ϕ with symbol ϕ is the operator on L 2 a (Ω) defined by More general results concerning which operators, not necessarily Toeplitz, commute with an analytic Hardy space Toeplitz operator are due to Thompson ([10] and [11]) and Cowen [6]. On the Bergman space, the situation is more complicated. The Brown-Halmos result mentioned above fails. For example, if Ω is the unit disk, then any two Toeplitz operators whose symbols are radial functions commute (proof: an easy calculation shows that every Toeplitz operator with radial symbol has a diagonal matrix with respect to the usual orthonormal basis; any two diagonal matrices commute).

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“…It follows from [20,Theorem 4.4.9] that a real-valued function u defined on a simply connected domain G is pluriharmonic if and only if u is the real part of a holomorphic function on G. Clearly, a mapping f : B n → C is pluriharmonic if and only if f has a representation f = h + g, where g and h are holomorphic. We refer to [4,6,13,15,20] for the definition and further details on pluriharmonic mappings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%