On the essential norms of Toeplitz operators
on abstract Hardy spaces built upon Banach function spaces

For a Banach space $\mathcal{X}$, let $\mathcal{B}(\mathcal{X})$ denote the Banach algebra of bounded linear operators on $\mathcal{X}$ and let $\mathcal{K}(\mathcal{X})$ be the closed two-sided ideal of $\mathcal{B}(\mathcal{X})$ consisting of all compact linear operators on $\mathcal{X}$. The norm of an operator $A\in\mathcal{B}(\mathcal{X})$ is denoted by $\|A\|_{\mathcal{B}(\mathcal{X})}$. The essential norm of $A\in\mathcal{B}(\mathcal{X})$ is defined as follows:

For a function $f\in L^{1}$ on the unit circle $\mathbb{T}:=\left\{z\in\mathbb{C}:\ |z|=1\right\}$ equipped with the Lebesgue measure $m$ normalised so that $m(\mathbb{T})=1$, let

be the Fourier coefficients of $f$. Let $X$ be a Banach function space on the unit circle $\mathbb{T}$. We postpone the definition of this notion until Section 2.1. Here we only mention that the class of Banach function spaces is very reach, it includes all Lebesgue spaces $L^{p}$, $1\leq p\leq\infty$, Orlicz spaces $L^{\Phi}$ (see, e.g., [1, Ch. 4, Section 8]), and Lorentz spaces $L^{p,q}$, $1<p<\infty$, $1\leq q\leq\infty$ (see, e.g., [1, Ch. 4, Section 4]). Moreover, all mentioned above spaces are rearrangement-invariant (see Section 2.2 for their definition).

Let

denote the abstract Hardy space built upon the space $X$. In the case $X=L^{p}$, where $1\leq p\leq\infty$, we will use the standard notation $H^{p}:=H[L^{p}]$. Consider the operators $S$ and $P$, defined for a function $f\in L^{1}$ and an a.e. point $t\in\mathbb{T}$ by

respectively, where the integral is understood in the Cauchy principal value sense. The operator $S$ is called the Cauchy singular integral operator and the operator $P$ is called the Riesz projection. Assume that the Riesz projection is bounded on $X$. For $a\in L^{\infty}$, the Toeplitz operator with symbol $a$ is defined by

It is clear that $T(a)\in\mathcal{B}(H[X])$ and

Let $C$ denote the Banach space of all complex-valued continuous functions on $\mathbb{T}$ with the supremum norm and let

In 1967, Sarason observed that $C+H^{\infty}$ is a closed subalgebra of $L^{\infty}$ (see, e.g., [4, Ch. IX, Theorem 2.2] for the proof of this fact).

Let $1<p<\infty$ and $a\in L^{\infty}$. It follows from [3, Theorem 2.30] and (1.1) that

I. Gohberg and N. Krupnik [5, Theorem 6] proved that $\|P\|_{\mathcal{B}(L^{p}),\mathrm{e}}\geq 1/\sin(\pi/p)$ and conjectured that $\|P\|_{\mathcal{B}(L^{p})}=1/\sin(\pi/p)$. This conjecture was confirmed by B. Hollenbeck and I. Verbitsky in [6]. Thus

A. Böttcher, N. Krupnik, and B. Silbermann [2, Section 7.6] asked whether the essential norm of Toeplitz operators $T(a)$ with $a\in C$ acting on the Hardy spaces $H^{p}$ is independent of $p\in(1,\infty)$. The second author answered this question in the negative [12]. More precisely, it was shown that

Nevertheless, the following estimates for $\|T(a)\|_{\mathcal{B}(H^{p}),\mathrm{e}}$ were obtained for $1<p<\infty$ and $a\in C+H^{\infty}$:

(see (1.2) and [12, Theorem 4.1]).

We will use the following notation:

The following result extends (1.3) to the class of rearrangement-invariant Banach function spaces.

The implication (f) $\Longrightarrow$ (a) follows from inequalities (1.2), which become equalities for $p=2$. The implications (a) $\Longrightarrow$ (b) $\Longrightarrow$ (c) are trivial. The equivalences (d) $\Longleftrightarrow$ (e) $\Longleftrightarrow$ (f) were proved in [10, Theorems 1.1–1.2] for arbitrary (not necessarily rearrangement-invariant) Banach function spaces $X$. The equality

was proved in [8, Theorem 1.2] for rearrangement-invariant Banach function spaces $X$, which gives the equivalence (c) $\Longleftrightarrow$ (d) and completes the proof of Theorem 1.1.

Böttcher, Krupnik, and Silbermann [2, p. 472] provided an argument allowing to show directly that (c) $\Longrightarrow$ (b) in the case of classical Hardy spaces $H^{p}$, $1<p<\infty$. The aim of this paper is to show that their reasoning can be extended to the case of arbitrary Banach function spaces (not necessarily rearrangement-invariant) on which the Riesz projection $P$ is bounded. Our main result is the following.

The paper is organised as follows. In Section 2, we recall the definition of the class of Banach function spaces and of its distinguished subclasss of rearrangement-invariant Banach function spaces. In Section 3, we prove that the Toeplitz operators $T(\mathbf{e}_{-n}h)$ with $n\in\mathbb{Z}_{+}:=\{0,1,2,\dots\}$ and $h\in H^{\infty}$ are bounded on $H[X]$. Further, we show that $T(\mathbf{e}_{-1})T(\mathbf{e}_{-n}h)=T(\mathbf{e}_{-n-1}h)$ for $n\in\mathbb{Z}_{+}$ and $h\in H^{\infty}$ on the space $H[X]$. Although our main results have been obtained under the assumption that $P$ is bounded on $X$, we do not make this assumption in Section 3 as we believe that this more general case is of an independent interest. By using the results of Section 3, we prove Theorem 1.2 in Section 4.

This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/ 00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications).