arXiv subjects

João R. Carmo

Publications and source records attributed to João R. Carmo.

3 recordsLinked to original sources

arXiv2026

Li-Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk

We study Li--Yorke and mean Li--Yorke chaos for weighted composition operators $C_{w,φ}$ on Banach spaces of analytic functions on the unit disk $\mathbb{D}$. Under natural conditions on the space, we show that $C_{w,φ}$ is (densely) Li--Yorke chaotic if and only if it is not power-bounded, and (densely) mean Li--Yorke chaotic if and only if it is not absolutely Cesàro bounded. These results are applied to Hardy spaces $H^p(\mathbb{D})$, $1 \le p \le \infty$, and weighted Bergman spaces $A^p_β(\mathbb{D})$, $-1 < β< \infty$ and $1 < p < \infty$.

math.FA↗

arXiv2023

Minimal invariant subspaces for an affine composition operator

The composition operator $C_{ϕ_a}f=f\circϕ_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for $C_{ϕ_a}$ is one-dimensional. These minimal invariant subspaces are always singly-generated $ K_f := \overline{\mathrm{span} \{f, C_{ϕ_a}f, C^2_{ϕ_a}f, \ldots \}}$ for some $f\in H^2(\mathbb{D})$. In this article we characterize the minimal $K_f$ when $f$ has a nonzero limit at the point $1$ or if its derivative $f'$ is bounded near $1$. We also consider the role of the zero set of $f$ in determining $K_f$. Finally we prove a result linking universality in the sense of Rota with cyclicity.

math.FA↗

arXiv2021

Universal composition operators

A Hilbert space operator $U$ is called universal (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for $U$ are one dimensional. In this article we characterize all linear fractional composition operators $C_ϕ f=f\circϕ$ that have universal translates on both the classical Hardy spaces $H^2(\mathbb{C}_+)$ and $H^2(\mathbb{D})$ of the half-plane and the unit disk respectively. The surprising new example is the composition operator on $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+(1-a)$ for $0<a<1$. This leads to strong characterizations of minimal invariant subspaces and eigenvectors of $C_{ϕ_a}$ and offers an alternative approach to the ISP.

math.FA↗