English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

The maximal ideal space of 𝐻^{∞}(𝔻) with respect to the Hadamard product