Universal meromorphic functions

Chan, Kit C.

doi:10.1080/17476930108815418

Cited by 10 publications

(8 citation statements)

References 3 publications

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“…In Theorem 2.1 below, we construct a universal element without the assumption that the semigroup X is a topological semigroup, nor the assumption that X possess an identity element. Then in Corollary 2.2, we show how those two assumptions reduce Theorem 2.1 to the format of the universality criterion obtained in [6] and [23].…”

Section: Main Results and Outline Of Dissertationmentioning

confidence: 86%

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A constructive approach to the Universality Criterion for semi-groups

Chan¹,

Walmsley²

2019

Mathematical Proceedings of the Royal Irish Academy

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We prove a generalization of the well-known Universality Criterion in the setting of continuous homomorphisms acting on a separable, complete, metrizable topological semigroup by constructing a particular universal element. Consequently, we simplify the proof of a 1955 classical result of Heins on the existence of universal Blaschke products on the unit ball of H ∞ (D), and the proofs of a number of other related results, by unifying them in the semigroup setting. Motivated by our generalization, we provide new applications such as the homomorphisms of conjugation on the unit ball of the operator algebra of c 0 and p , where 1 ≤ p < ∞, and also the homomorphisms of composition on the closed unit ball Ball L ∞ [0, 1] of L ∞ [0, 1] with the weak-star topology, as well as on the semigroup L 1 (R) ∩ Ball L ∞ (R). iv ACKNOWLEDGMENTS I would like to thank Dr. Chan for his unwavering patience and keen guidance. His relaxed nature and abundant kindness made the whole process truly enjoyable. His prowess as a speaker is parallel to none, as he can make even the most complicated argument understandable. He is a true master, and none of this would be possible without him.I would also like to thank Dr. Seubert, who has had a profound impact on my teaching philosophy and practice. His passion for teaching students to think and his ability to convey the bigger picture are contagious. I hope to emulate his passion in the classroom and show my future students that what we are training them to do matters in the real world.I have made some great friends along the way. Shout outs, in no particular order, to Jake

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Section: Main Results and Outline Of Dissertationmentioning

confidence: 86%

“…In this section we construct a universal element in a separable, complete, metrizable topological semigroup X for a sequence of continuous homomorphisms on X satisfying the well-known universality criterion which was first obtained by Chan [6] and later by Moothathu [23].…”

Section: Main Results and Outline Of Dissertationmentioning

confidence: 99%

A constructive approach to the Universality Criterion for semi-groups

Chan¹,

Walmsley²

2019

Mathematical Proceedings of the Royal Irish Academy

Self Cite

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show abstract

“…Lastly, we conclude our discussion with some other universality results for C(R) and L 1 (R). In this section we construct a universal element in a separable, complete, metrizable topological semigroup X for a sequence of continuous homomorphisms on X satisfying the well-known universality criterion which was first obtained by Chan [6] and later by Moothathu [23].…”

Section: Main Results and Outline Of Dissertationmentioning

confidence: 99%

A constructive approach to the Universality Criterion for semi-groups

Chan¹

2019

Mathematical Proceedings of the Royal Irish Academy

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We prove a generalization of the well-known Universality Criterion in the setting of continuous homomorphisms acting on a separable, complete, metrizable topological semigroup by constructing a particular universal element. Consequently, we simplify the proof of a 1955 classical result of Heins on the existence of universal Blaschke products on the unit ball of H ∞ (D), and the proofs of a number of other related results, by unifying them in the semigroup setting. Motivated by our generalization, we provide new applications such as the homomorphisms of conjugation on the unit ball of the operator algebra of c 0 and p , where 1 ≤ p < ∞, and also the homomorphisms of composition on the closed unit ball Ball L ∞ [0, 1] of L ∞ [0, 1] with the weak-star topology, as well as on the semigroup L 1 (R) ∩ Ball L ∞ (R). iv ACKNOWLEDGMENTS I would like to thank Dr. Chan for his unwavering patience and keen guidance. His relaxed nature and abundant kindness made the whole process truly enjoyable. His prowess as a speaker is parallel to none, as he can make even the most complicated argument understandable. He is a true master, and none of this would be possible without him. I would also like to thank Dr. Seubert, who has had a profound impact on my teaching philosophy and practice. His passion for teaching students to think and his ability to convey the bigger picture are contagious. I hope to emulate his passion in the classroom and show my future students that what we are training them to do matters in the real world. I have made some great friends along the way. Shout outs, in no particular order, to Jake

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“…The final result of this section is concerned with universal locally univalent meromorphic functions. Chan [8] has shown that there exists a meromorphic function f ∈ M(C) such that the set T f := {f (· + n) : n ∈ N} is dense in M(Ω) for every domain Ω ⊆ C.…”

Section: Universal Locally Univalent Functionsmentioning

confidence: 99%

“…(3) (Universal meromorphic functions) There are Birkhoff-type universality results for meromorphic functions (see e.g. [8]).…”

Section: Introductionmentioning

confidence: 99%