A characterization of maximal ideals in the Fr\'{e}chet algebras of holomorphic functions $F^p$ $(1<p<\infty$

TL;DR

This paper characterizes the structure of maximal ideals and multiplicative linear functionals in the Fréchet algebras of holomorphic functions $F^p$, providing a complete description of their algebraic and functional properties.

Contribution

It offers a complete characterization of closed maximal ideals and multiplicative linear functionals in the $F^p$ algebras, advancing understanding of their algebraic structure.

Findings

Complete description of closed maximal ideals in $F^p$

Characterization of multiplicative linear functionals on $F^p$

Connection between $F^p$ and Privalov space $N^p$

Abstract

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $\lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms $\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fr\'{e}chet algebra. Notice that for each $p>1$, $F^p$ is the Fr\'{e}chet envelope of the Privalov space $N^p$. In this paper we study the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). In particular, we give a complete characterization of closed maximal ideals in $F^p$. Moreover, we characterize multiplicative linear functionals on $F^p$.

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Full text

Notice that for each $p>1$, $F^{p}$ is the Fréchet envelope of the Privalov space $N^{p}$. In this paper we study the structure of maximal ideals in the algebras $F^{p}$ ($1<p<\infty$). In particular, we give a complete characterization of closed maximal ideals in $F^{p}$. Moreover, we characterize multiplicative linear functionals on $F^{p}$.

†† Mathematics Subject Classification (2010). 30H05, 46J15, 46J20.

1. Introduction, Preliminaries and Results

Let $\mathbb{D}$ denote the open unit disk in the complex plane and let $\mathbb{T}$ denote the boundary of $\mathbb{D}$. Let $L^{q}(\mathbb{T})$ $(0<q\leq\infty)$ be the familiar Lebesgue space on the unit circle $\mathbb{T}$.

The Privalov class $N^{p}$ $(1<p<\infty)$ is defined as the set of all holomorphic functions $f$ $f$ on $\mathbb{D}$ such that

[TABLE]

holds, where $\log^{+}|a|=\max\{\log|a|,0\}$. These classes were firstly considered by Privalov in [1, pages 93–10], where $N^{p}$ is denoted as $A_{q}$.

Notice that for $p=1$, the condition (1) defines the Nevanlinna class $N$ of holomorphic functions in $\mathbb{D}$. Recall that the Smirnov class $N^{+}$ is the set of all functions $f$ holomorphic on $\mathbb{D}$ such that

[TABLE]

where $f^{*}$ is the boundary function of $f$ on $\mathbb{T}$; that is,

[TABLE]

is the radial limit of $f$ which exists for almost every $e^{i\theta}\in\mathbb{T}$. We denote by $H^{q}$ $(0<q\leq\infty)$ the classical Hardy space on $\mathbb{D}$.

It is known (see [2, 3, 4]) that the following inclusion relations hold:

[TABLE]

where the above containment relations are proper.

The study of the spaces $N^{p}$ $(1<p<\infty)$ was continued in 1977 by M. Stoll [5] (with the notation $(\log^{+}H)^{\alpha}$ in [5]). Further, the topological and functional properties of these spaces have been studied by several authors (see [2], [6], [7], [8] and [9]–[23]).

M. Stoll [5, Theorem 4.2] proved that for each $p>1$ the space $N^{p}$ (with the noatation $(\log^{+}H)^{\alpha}$ in [5]) equipped with the topology given by the metric $d_{p}$ defined by

[TABLE]

becomes an $F$-algebra, that is, $N^{p}$ is an $F$-space (a complete metrizable topological vector space with the invariant metric) in which multiplication is continuous.

Recall that the function $d_{1}=d$ defined on the Smirnov class $N^{+}$ by (5) with $p=1$ induces the metric topology on $N^{+}$. N. Yanagihara [24] showed that under this topology, $N^{+}$ is an $F$-space.

In connection with the spaces $N^{p}$ $(1<p<\infty)$, Stoll [5] (see also [6] and [18, Section 3]) also studied the spaces $F^{q}$ $(0<q<\infty)$ (with the notation $F_{1/q}$ in [5]), consisting of those functions $f$ holomorphic on $\mathbb{D}$ such that

[TABLE]

where

[TABLE]

Here, as always in the sequel, we will need some Stoll’s results concerning the spaces $F^{q}$ only with $1<q<\infty$, and hence, we will assume that $q=p>1$ be any fixed number.

Theorem 1 (see [5, Theorem 2.2]). Suppose that $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is a holomorphic function on $\mathbb{D}$. Then the following statements are equivalent:

• (a)

$f\in F^{p}$;

• (b)

there exists a sequence $\{c_{n}\}_{n}$ of positive real numbers with $c_{n}\to 0$ such that

[TABLE]

• (c)

for any $c>0$,

[TABLE]

Remark 2. Notice that in view of Theorem 1 ((a)$\Leftrightarrow$(c)), by (10) it is well defined the family of seminorms $\left\{\|\cdot\|_{p,c}\right\}_{c>0}$ on $F^{p}$.

Recall that a locally convex $F$-space is called a Fréchet space, and a Fréchet algebra is a Fréchet space that is an algebra in which multiplication is continuous. Stoll [5] also proved the following result.

Theorem 3 (see [5, Theorem 3.2]). The space $F^{q}$ $(0<q<\infty)$ equipped with the topology given by the family of seminorms $\left\{\|\cdot\|_{q,c}\right\}_{c>0}$ defined for $f\in F^{q}$ as

[TABLE]

is a countably normed Fréchet algebra.

For our purposes, we will need the following result which characterizes the topological dual of the space $F^{p}$.

Theorem 4 (see [5, Theorem 3.3]). If $\gamma$ is a continuous linear functional on $F^{p}$, then there exists a sequence $\{\gamma_{n}\}_{n}$ of complex numbers with

[TABLE]

such that

[TABLE]

where $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in F^{p}$, with convergence being absolute. Conversely, if $\{\gamma_{n}\}_{n}$ is a sequence of complex numbers for which

[TABLE]

then (13) defines a continuous linear functional on $F^{p}$.

Notice that the Privalov space $N^{p}$ $(1<p<\infty)$ is not locally convex (see [6, Theorem 4.2] and [14, Corollary]), and hence, $N^{p}$ is properly contained in $F^{p}$. Moreover, $N^{p}$ is not locally bounded (see [19, Theorem 1.1]). Moreover, Stoll showed ([5, Theorem 4.3]) that for each $p>1$ $N^{p}$ is a dense subspace of $F^{p}$ and the topology on $F^{p}$ defined by the family of seminorms (10) is weaker than the topology on $N^{p}$ given by the metric $d_{p}$ defined by (5). Furthermore, Eoff showed [6, Theorem 4.2, the case $p>1$] that $F^{p}$ is the Fréchet envelope of $N^{p}$. For more information on Fréchet envelope, see [25, Theorem 1], [22, Section 1] and [26, Corollary 22.3, p. 210].

Remark 5. For $p=1$, the space $F_{1}$ has been denoted by $F^{+}$ and has been studied by N. Yanagihara in [27, 24]. It was shown in [27, 24] that $F^{+}$ is actually the containing Fréchet space for $N^{+}$, i.e., $N^{+}$ with the initial topology embeds densely into $F^{+}$, under the natural inclusion, and $F^{+}$ and the Smirnov class $N^{+}$ have the same topological duals.

Observe that the space $F^{p}$ topologised by the family of seminorms $\left\{\|\cdot\|_{p,c}\right\}_{c>0}$ given by (10) is metrizable by the metric $\lambda_{p}$ defined as $\lambda_{p}(f,g)=\sum_{n=1}^{\infty}2^{-n}\frac{\|f-g\|_{p,1/n^{p/(p+1)}}}{1+\|f-g\|_{p,1/n^{p/(p+1)}}}$ with $f,g\in F^{p}.$

Since Privalov space $N^{p}$ and its Fréchet envelope $F^{p}$ $(1<p<\infty)$ are algebras, they can be also considered as rings with respect to the usual ring’s operations addition and multiplication. Notice that these two operations are continuous on $N^{p}$ and $F^{p}$ because the spaces $N^{p}$ and $F^{p}$ become $F$-algebras.

Motivated by several results on the ideal structure of some spaces of holomorphic functions given in [28] [2], [12] and [29]-[35], related investigations for the spaces $N^{p}$ $(1<p<\infty)$ and their Fréchet envelopes were given in [2], [9], [12], [36], [18] and [23]. Note that a survey of these results was given in [37]. The $N^{p}$-analogue of the famous Beurling’s theorem for the Hardy spaces $H^{q}$ $(0<q<\infty)$ [29] was proved in [36]. Moreover, it was proved in [9, Theorem B]) that $N^{p}$ $(1<p<\infty)$ is a ring of Nevanlinna–Smirnov type in the sense of Mortini. The structure of closed weakly dense ideals in $N^{p}$ was established in [18]. The ideal structure of $N^{p}$ and the multiplicative linear functionals on $N^{p}$ were studied in [2] and [23, Theorem ]. These results are similar to those obtained by Roberts and Stoll [30] for the Smirnov class $N^{+}$.

Motivated by results of Roberts and Stoll given in [31, Section 2] concerning a characterization of multiplicative linear functionals on $F^{+}$ and closed maximal ideals in $F^{+}$, in this paper we prove the analogous results for the spaces $F^{p}$ $(1<p<\infty)$ given by Proposition 5, Proposition 6, Theorem 7 and Theorem 8.

Proposition 5. Let $\lambda\in\mathbb{D}$ and let $\gamma_{\lambda}$ be a functional on $F^{p}$ defined as

[TABLE]

for every $f\in F^{p}$. Then $\gamma_{\lambda}$ is a continuous multiplicative linear functional on $F^{p}$.

For $\lambda\in\mathbb{D}$, we define

[TABLE]

Proposition 6. The set ${\mathcal{M}}_{\lambda}$ defined by $(15)$ is a closed maximal ideal in $F^{p}$ for each $\lambda\in\mathbb{D}$.

Theorem 7. Let $\gamma$ be a nontrivial multiplicative linear functional on $F^{p}$. Then there exists $\lambda\in\mathbb{D}$ such that

[TABLE]

for every $f\in F^{p}$. Moreover, $\gamma$ is a continuous map.

Theorem 8. Let $p>1$ and let ${\mathcal{M}}$ be a closed maximal ideal in $F^{p}$. Then there exists $\lambda\in\mathbb{D}$ such that ${\mathcal{M}}={\mathcal{M}}_{\lambda}$.

2. Proof of the results

Proof of Proposition 5. Clearly, for each $\lambda\in\mathbb{D}$ $\gamma_{\lambda}$ is a multiplicative linear functional on the space $F^{p}$. In order to show that $\gamma_{\lambda}$ is a continuous functional on $F^{p}$, note that for any function $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in F^{p}$ ($z\in\mathbb{D}$) we have

[TABLE]

Clearly, the sequence $\{\gamma_{n}\}$ defined as $\gamma_{n}=\lambda^{n}$ ($n=0,1,2\ldots$) satisfies the asymptotic condition (11) of Theorem 4. This together with the equality (17) implies that $\gamma_{\lambda}$ is a continuous functional on $F^{p}$, and the proof is completed. $\square$

Proof of Proposition 6. Notice that in view of (15), ${\mathcal{M}}_{\lambda}$ is the kernel of the functional $\gamma_{\lambda}$ defined on $F^{p}$ by (14). From this and the fact that by Proposition 5, $\gamma_{\lambda}$ is a continuous multiplicative linear functional on the space $F^{p}$, we conclude that ${\mathcal{M}}_{\lambda}$ is a closed maximal ideal in $F^{p}$. $\square$

Proof of Theorem 7. If we take $\gamma(z)=\lambda$, then $\gamma(z-\lambda)=0$. If we suppose that $\lambda\notin\mathbb{D}$, then $z\mapsto 1/(z-\lambda)$ ($z\in\mathbb{D}$) is a bounded function on the closed unit disk $\overline{\mathbb{D}}:|z|\leq 1$. Therefore, $z\mapsto z-\lambda$ ($z\in\mathbb{D}$) is an invertible element of the algebra $F^{p}$. If $f$ is any invertible element in $F^{p}$, then $1=\gamma(1)=\gamma(f)\gamma(f^{-1})$, and thus, $\gamma(f)\not=0$. Especially, we have $\gamma(z-\lambda)\not=0$. A contradiction, and hence, it must be $\lambda\in\mathbb{D}$. Then consider the set

[TABLE]

For each $\lambda\in\mathbb{D}$, let ${\mathcal{M}}_{\lambda}$ be a set defined by (15). Then obviously, $(z-\lambda)F^{p}\subset{\mathcal{M}}_{\lambda}$. Moreover, if $f\in{\mathcal{M}}_{\lambda}$, then by (6) and (7) easily follows that $f$ can be expressed as a product $f(z)=(z-\lambda)g(z)$ with $g\in F^{p}$. Therefore,

[TABLE]

whence it follows that

[TABLE]

where $\ker\gamma$ denotes the kernel of the functional $\gamma$. By Proposition 6, ${\mathcal{M}}_{\lambda}$ is a closed maximal ideal in $F^{p}$. This together with the inclusion relation (20) implies that ${\mathcal{M}}_{\lambda}=\ker\gamma$. Moreover, $\gamma(f)=f(\lambda)$ for all $f\in F^{p}$ and $\gamma$ is continuous on $F^{p}$ by Proposition 5. This completes the proof of the theorem. $\square$

Proof of Theorem 8. We proceed as in [32, Theorem 2]. If we set $X=F^{p}/{\mathcal{M}}$, then in the terminology of Arens [38], $X$ is complete, metrizable, convex complex topological division algebra. Therefore, by [38], $X\cong\mathbb{C}$. Thus, there exists a multiplicative linear functional $\gamma$ on $F^{p}$ such that ${\mathcal{M}}=\ker\gamma$. Then by Theorem 7, ${\mathcal{M}}={\mathcal{M}}_{\lambda}$ for some $\lambda\in\mathbb{D}$, as asserted. $\square$