arXiv subjects

Abdullah Bin Abu Baker

Publications and source records attributed to Abdullah Bin Abu Baker.

5 recordsLinked to original sources

arXiv2026

Generalized tri-circular projections on some vector-valued spaces of analytic functions

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 $E$-valued bounded analytic functions for a Banach space $E$ with trivial multiplier algebra; and the little Bloch spaces $\mathcal{B}_0(\mathbb{D}, E)$ and $\mathcal{B}_{\ast}(\mathbb{D}, E)$, for $E$ smooth, strictly convex, and reflexive.

math.FA↗

arXiv2026

Structure of Generalized bi-circular idempotents and isometric reflections on $C^1[0,1]$

Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \|f\|_σ = |f(0)| +\|f'\|_\infty$. Let $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $λ_1P_1 + λ_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $λ_1, λ_2$, if and only if either $λ_1 + λ_2 = 0$, or $λ_1P_1 + λ_2P_2$ is an isometry for all such complex numbers $λ_1, λ_2$. In the former case, the collection $\{P_1, P_2\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.

math.FA↗

arXiv2026

Generalized tri-circular projections on some spaces of analytic functions

In this paper, we obtain the structure of a class of contractive projections, known as generalized tri-circular projections, on some functional Banach spaces on the open unit disk $\mathbb{D}$, which includes Hardy, Bergman and Bloch spaces. We then consider the space $S^p_{\mathcal{K}}$ of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$ such that $f' \in H^p(\mathcal{K})$ and give complete description of generalized tri-circular projections on this space. Here, $\mathcal{K}$ is a complex separable Hilbert space, and $H^p(\mathcal{K})$ denotes the Hardy space of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$.

math.FA↗

arXiv2026

On Nonlinear Idempotents and Metric Projections in Normed Spaces

Let $X$ be a normed linear space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$. In this paper, we introduce some new classes of idempotent maps (linearity is not assumed) on $X$ and study their properties. A well-known example of an idempotent map is the metric projection $P_K$ onto a Chebyshev subset $K$ of $X$. We provide some necessary conditions for the map $I- P_K$ to be idempotent, where $I$ denotes the identity operator on $X$.

math.FA↗

arXiv2026

Generalized idempotents on the space of analytic functions with bounded derivatives

Let $X$ be a complex normed space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$. A collection $\mathcal{C} = \{P_1, P_2\}$ of nonzero distinct orthogonal ($P_1P_2 = P_2P_1 = 0$) idempotent maps on $X$ is said to be a family of generalized bi-circular idempotents if there exist distinct unit modulus complex numbers $λ_1, λ_2$ such that $P_1 + P_2 = I$ (identity operator on $X$) and $λ_1P_1 + λ_2P_2$ is a surjective isometry on $X$. This generalizes the notion of generalized bi-circular projections on Banach spaces introduced by Fošner, Ilišević and Li \cite{MDC} to nonlinear maps. In this paper, we describe the structure of generalized bi-circular idempotents over the space of analytic functions on the open unit disk with bounded derivatives.

math.FA↗