We identify properties of a Banach space $\mathcal{B}$ of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane ensuring that a multiplication operator $M_\psi: \mathcal{B} \to\mathcal{B}$ is Fredholm if and only if its symbol $\psi$ is bounded away from $0$ near $\partial \mathbb{D}$. The properties we identify are shared by a wide variety of much-studied spaces, including the Hardy spaces $H^p(\mathbb{D})$, weighted Bergman spaces $A^p_\omega(\mathbb{D})$, Hardy-Sobolev spaces $H^2_\beta(\mathbb{D})$, the spaces $S_j^p(\mathbb{D})$ of functions having $j$-th derivative in $H^p(\mathbb{D})$, and the disk algebra $A$. Thus, as a corollary, our work characterizes Fredholm multiplication operators on these spaces. In addition, we describe the closed, finite-codimensional subspaces of $\mathcal{B}$ that are invariant under $M_z: \mathcal{B} \to \mathcal{B}$ in terms of the zeros that the functions in such subspaces have in common. We discuss connections between these subspaces and the problem of characterizing the Fredholm multiplication operators on $\mathcal{B}$, and we prove that $M_z$ restricted to such a subspace is always cyclic, with a polynomial cyclic vector having degree equal to the codimension of the subspace.
In this paper, we characterize generalized tri-circular projections on the following three vector-valued spaces of analytic functions on the open unit disk $\mathbb{D}$: the Hardy space $H^p(\mathcal{K})$ for a complex separable Hilbert space $\mathcal{K}$ and $1 \le p<\infty,p \neq 2$; the space $H^\infty(E)$ of $E$-v...
H. Kumar, H. Kumar, Abdullah Bin Abu Baker· 0 citations
We study the Banach space and operator factorization structure of Janson-Sobolev and Janson-Sobolev-Hardy spaces. This new class of martingale spaces is determined by a $q$-adic filtration, a subspace $V\subset \mathbb{R}_0^{l\times q}$, and a rearrangement invariant function space $X$. Our main result shows that, for...
Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $...
D. Norrbo, J. A. Pel'aez, Fang-Lei Wu· 2 citations· ⚡1
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
Let $(\mathcal{X},d,\mu)$ be a doubling metric measure space, $X$ a ball quasi-Banach function space on $\mathcal{X}$, and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernel satisfies a Gaussian upper bound. In this article, we study the local Hardy space $h_{X,L}(\mathcal{X})$ associated...
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
