arXiv subjects

Marcos S. Ferreira

Publications and source records attributed to Marcos S. Ferreira.

5 recordsLinked to original sources

arXiv2026

Minimality of $T_z^*$-Invariant Subspaces of $H^2(\mathbb D^2)$

In this paper, we establish necessary conditions for the minimality of infinite-dimensional $T_z^*$-invariant subspaces of $H^2(\mathbb D^2)$. More precisely, we prove that if $M\subset H^2(\mathbb D^2)$ is an infinite-dimensional, minimal $T_z^*$-invariant subspace, then, for every $n\in\mathbb N_0$, either $\overline{P_n(M)}=H^2(\mathbb D)$ or $\overline{P_n(M)}^\perp =ϕ_{a,θ}H^2(\mathbb D)$ where $ϕ_{a,θ}$ is an atomic singular inner function and $P_n$ denotes the orthogonal projection of $H^2(\mathbb D^2)$ onto $H^2(z)w^n$. As $T_z^*$ is a universal operator on $H^2(\mathbb D^2)$, this result reduces the Invariant Subspace Problem to the analysis of these two alternatives for infinite-dimensional minimal $T_z^*$-invariant subspaces.

math.FA↗

arXiv2023

On the invariant subspace problem via universal Toeplitz operators on the Hardy space $H^{2}(\mathbb{D}^{2})$

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this paper, we obtain a nontrivial invariant subspace of $T^{*}_φ|_{M}$, where $T_φ$ is the Toeplitz operator on the Hardy space over the bidisk $H^{2}(\mathbb{D}^{2})$ induced by the symbol $φ\in H^{\infty}(\mathbb{D})$ and $M$ is a $T_φ^{*}$-invariant subspace. We use this fact to get sufficient conditions for the ISP.

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arXiv2022

A characterization for complex symmetric Toeplitz operators

In this paper we use orthonormal basis for the Hardy space $H^{2}(\mathbb{T})$, formed by rational functions, to characterize complex symmetric Toeplitz operators on $H^{2}(\mathbb{T})$. As a result, we get examples of these operators whose symbols are non-trigonometric functions.

math.FA↗

arXiv2022

Conjugations on the Hardy space $H^{2}$

A conjugation $C$ on a separable complex Hilbert space $\mathcal H$ is an antilinear operator that is isometric and involutive. In this notes, we characterize all conjugations on the Hardy-Hilbert space $H^{2}$ over the disk. In addition, we characterize complex symmetric Toeplitz operators with a special type of these conjugations.

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arXiv2019

Some Notes on Complex Symmetric Operators

In this paper we show that every conjugation $C$ on the Hardy-Hilbert space $H^{2}$ is of type $C=T^{*}C_{1}T$, where $T$ is an unitary operator and $C_{1}f\left(z\right)=\overline{f\left(\overline{z}\right)}$, with $f\in H^{2}$. In the sequence, we extend this result for all separable Hilbert space $\mathcal H$ and we prove some properties of complex symmetry on $\mathcal H$. Finally, we prove some relations of complex symmetry between the operators $T$ and $\left|T\right|$, where $T =U\left|T\right|$ is the polar decomposition of bounded operator $T\in\mathcal L\left(\mathcal H\right)$ on the separable Hilbert space $\mathcal H$.

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