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F. Javier González-Doña

Publications and source records attributed to F. Javier González-Doña.

8 recordsLinked to original sources
arXiv2026

On frequencies of parabolic Koenigs domains

Let $(φ_t)_{t\geq 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $Ω= h(\mathbb{D})$. We study the point spectrum $σ_p(Δ\mid_{H^p})$ of $Δ$, the infinitesimal generator of the $C_0$-semigroup $(C_{φ_t})_{t\geq 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $Ω$. That is, those $λ\in \mathbb{C}$ such that $e^{λh} \in H^p$. We derive containment relations for $σ_p(Δ\mid_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $Ω$ and on careful estimates of the harmonic measure of some boundary subsets of $Ω$. We conclude with some consequences regarding the spectrum of the composition operators $(C_{φ_t})_{t\geq 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups.

math.CV↗
arXiv2026

Spectral cuts and unconventional functional calculi

In this work, we prove that linear bounded operators $T$ on a Banach space $X$ allowing spectral cuts along rectifiable Jordan curves meeting their spectrum are related to classes of operators admitting an unconventional functional calculus. We identify several such classes and address the consequences regarding the existence of non-trivial closed invariant subspaces, extending previous results of Chalendar. Furthermore, we establish that every operator belonging to a broad subclass of compact perturbations of diagonalizable normal operators on separable Hilbert spaces, namely, trace-class perturbations, possesses an unconventional functional calculus and is super-decomposable, thereby extending earlier results obtained by the authors.

math.FA↗
arXiv2025

Local spectral theory for subordinated operators: the Cesàro operator and beyond

We study local spectral properties for subordinated operators arising from $C_0$-semigroups. Specifically, if $\mathcal{T}=(T_t)_{t\geq 0}$ is a $C_0$-semigroup acting boundedly on a complex Banach space and $$\mathcal{H}_ν= \int_{0}^{\infty} T_t\; dν(t)$$ is the subordinated operator associated to $\mathcal{T}$, where $ν$ is a sufficiently regular complex Borel measure supported on $[0,\infty)$, it is shown that $\mathcal{H}_ν$ does not enjoy the Single Valued Extension Property (SVEP) and has dense glocal spectral subspaces in terms of the spectrum of the generator of $\mathcal{T}$. Likewise, the adjoint $\mathcal{H}_ν^{\ast}$ has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator $\mathcal{C}$ acting on the Hardy spaces $H^p$ ($1<p<\infty$), it follows that the local spectrum of $\mathcal{C}$ at any non-zero $H^p$-function or the spectrum of the restriction of $\mathcal{C}$ to any of its non-trivial closed invariant subspaces coincides with the spectrum of $\mathcal{C}$. Finally, we characterize the local spectral properties of subordinated operators arising from hyperbolic semigroups of composition operators acting on $H^p$ ($1<p<\infty$), which will depend only on the geometry of the associated Koenigs domain.

math.FA↗
arXiv2024

On commutants of composition operators embedded into $C_0$-semigroups

Let $C_φ$ be a composition operator acting on the Hardy space of the unit disc $H^p$ ($1\leq p < \infty$), which is embedded in a $C_0$-semigroup of composition operators $\mathcal{T}=(C_{φ_t})_{t\geq 0}.$ We investigate whether the commutant or the bicommutant of $C_φ$, or the commutant of the semigroup $\mathcal{T}$, are isomorphic to subalgebras of continuous functions defined on a connected set. In particular, it allows us to derive results about the existence of non-trivial idempotents (and non-trivial orthogonal projections if $p=2$) lying in such sets. Our methods also provide results concerning the minimality of the commutant and the double commutant property, in the sense that they coincide with the closure in the weak operator topology of the unital algebra generated by the operator. Moreover, some consequences regarding the extended eigenvalues and the strong compactness of such operators are derived. This extends previous results of Lacruz, León-Saavedra, Petrovic and Rodríguez-Piazza, Fernández-Valles and Lacruz and Shapiro on linear fractional composition operators acting on $H^2$.

math.FA↗
arXiv2024

Universality arising from invertible weighted composition operators

A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its invariant subspaces. It turns out that characterizing the lattice of closed invariant subspaces of a universal operator is equivalent to solve the Invariant Subspace Problem for Hilbert spaces. In this paper, we consider invertible weighted hyperbolic composition operators and we prove the universality of the translations by eigenvalues of such operators, acting on Hardy and weighted Bergman spaces. Some consequences for the Banach space case are also discussed.

math.FA↗
arXiv2024

Dunford property for composition operators on $H^p$-spaces

The Dunford property $(C)$ for composition operators on $H^p$-spaces ($1<p<\infty$), as well as for their adjoints, is completely characterized within the class of those induced by linear fractional transformations of the unit disc. As a consequence, it is shown that the Dunford property is stable in such a class addressing a particular instance of a question posed by Laursen and Neumann.

math.FA↗
arXiv2024

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

Finite rank perturbations of diagonalizable normal operators acting boundedly on infinite dimensional, separable, complex Hilbert spaces are considered from the standpoint of view of the existence of invariant subspaces. In particular, if $T=D_Λ+u\otimes v$ is a rank-one perturbation of a diagonalizable normal operator $D_Λ$ with respect to a basis $\mathcal{E}=\{e_n\}_{n\geq 1}$ and the vectors $u$ and $v$ have Fourier coefficients $\{α_n\}_{n\geq 1}$ and $\{β_n\}_{n\geq 1}$ with respect to $\mathcal{E}$ respectively, it is shown that $T$ has non trivial closed invariant subspaces provided that either $u$ or $v$ have a Fourier coefficient which is zero or $u$ and $v$ have non zero Fourier coefficients and $$ \sum_{n\geq 1} |α_n|^2 \log \frac{1}{|α_n|} + |β_n|^2 \log \frac{1}{|β_n|} < \infty.$$ As a consequence, if $(p,q)\in (0,2]\times (0,2]$ are such $\sum_{n\geq 1} (|α_n|^p + |β_n|^q )< \infty,$ it is shown the existence of non trivial closed invariant subspaces of $T$ whenever $$(p,q)\in (0,2]\times (0,2]\setminus \{(2, r), (r, 2):\; r\in(1,2]\}.$$ Moreover, such operators $T$ have non trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity. Likewise, analogous results hold for finite rank perturbations of $D_Λ$. This improves considerably previous theorems of Foiaş, Jung, Ko and Pearcy, Fang and Xia and the authors on an open question explicitly posed by Pearcy in the seventies.

math.FA↗