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The maximal ideal space of ^{∞}() with respect to the Hadamard product
| Content Provider | Semantic Scholar |
|---|---|
| Author | Render, Hermann |
| Copyright Year | 1999 |
| Abstract | It is shown that the space of all regular maximal ideals in the Banach algebra H∞(D) with respect to the Hadamard product is isomorphic to N0. The multiplicative functionals are exactly the evaluations at the n-th Taylor coefficient. It is a consequence that for a given function f(z) = ∑∞ n=0 anz n in H∞(D) and for a function F (z) holomorphic in a neighborhood U of 0 with F (0) = 0 and an ∈ U for all n ∈ N0 the function g(z) = ∑∞ n=0 F (an)z n is in H∞(D). Introduction Let D := {z ∈ C : |z| < 1} be the open unit disk and let f(z) = ∑∞n=0 anz and g(z) = ∑∞ n=0 bnz n be power series on D. Then the Hadamard product of f and g is defined by f∗g(z) = ∑∞n=0 anbnz. The Hadamard product on the space H(D) of all holomorphic functions on D is continuous with respect to the topology of compact convergence. In [1] R. Brooks has shown that the space of all maximal ideals in the space H(D) is isomorphic to the Stone-Čech-compactification βN0 of N0 := N∪{0} and the multiplicative functionals on H(D) are given by the coefficient functionals δn : H(D) → C defined by δn(f) := an (where f(z) = ∑∞ n=0 anz n in |z| < 1 and n ∈ N0). In this note we discuss the subalgebra H∞(D) of all bounded holomorphic functions which has been considered for example in [3]. Our main result states that the non-trivial multiplicative functionals on H∞(D) are of the form δn, n ∈ N0 (as in the case of H(D)). In contrast to the algebra H(D) the space H∞(D) is even a Banach algebra with respect to the supremum norm which is denoted by ||f ||∞ for f ∈ H∞(D). It follows that the maximal modular ideals of H∞(D) are the kernels of the multiplicative functionals and therefore the space of all maximal modular ideals of H∞(D) is isomorphic to N0. Note that H(D) possesses a unit element γ(z) = 1 1−z = ∑∞ n=0 z n which is not in the subalgebra H∞(D). The results Let B be the space of all f(z) = ∑∞ n=0 anz n such that ∑∞ n=0 |an| < ∞; clearly, ‖f‖∞ ≤ ∑∞ n=0 |an|, so B ⊂ H∞(D). We note that if f = ∑∞ n=0 anz n ∈ H∞(D), then ∑∞ n=0 |an| = ‖f‖2 ≤ ‖f‖∞ < ∞, where ||f ||2 := √∑∞ n=0 |an|2. Hence for any f, g ∈ H∞(D), we have f ∗ g ∈ B, since ∑∞n=0 |anbn| ≤ ‖f‖2‖g‖2 by the Received by the editors March 27, 1997 and, in revised form, August 19, 1997. 1991 Mathematics Subject Classification. Primary 46J15; Secondary 30B10. |
| Starting Page | 1409 |
| Ending Page | 1411 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-99-04697-3 |
| Volume Number | 127 |
| Alternate Webpage(s) | https://www.ams.org/journals/proc/1999-127-05/S0002-9939-99-04697-3/S0002-9939-99-04697-3.pdf |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1999-127-05/S0002-9939-99-04697-3/S0002-9939-99-04697-3.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-99-04697-3 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
