arXiv subjects

Anurag Kumar Patel

Publications and source records attributed to Anurag Kumar Patel.

4 recordsLinked to original sources

arXiv2024

A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$

We give a characterization of zero divisors of the ring $C[a,b].$ Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra $C[a,b].$ We also characterize the zero divisors and topological divisors of zero in $\ell^\infty.$ Further, we show that zero is the only zero divisor in the disk algebra $\mathscr{A}(\mathbb{D})$ and that the class of singular elements in $\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of $\mathscr{A}(\mathbb{D})$ which are not zero divisors.

math.FA↗

arXiv2024

A class of weighted composition operators whose Range and Null spaces are complemented

In this paper, we prove that the null space of a weighted composition operator on $\ell_p~ (1 \leq p < \infty)$ is a complemented subspace. We also give a necessary and sufficient condition for a weighted composition operator on $\ell_p$ whose range space is of finite co-dimension. Thereafter, we characterize a class of weighted composition operators whose range space is a complemented subspace. Lastly, we characterize weighted composition operators on $\ell_p$ which are Fredholm operators.

math.FA↗

arXiv2024

Zero divisors and topological divisors of zero in certain Banach algebras

In this paper we prove that an element $f\in \mathcal{A}(\mathbb{D})$ is a topological divisor of zero(TDZ) if and only if there exists $z_0 \in \mathbb{T}$ such that $f(z_0)=0.$ We also give a characterization of TDZ in the Banach algebra $L^\infty(μ).$ Further, we prove that the multiplication operator $M_h$ is a TDZ in $\mathcal{B}(L^p(μ))~(1\leq p\leq\infty)$ if and only if $h$ is a TDZ in $L^\infty(μ).$ Subsequently, we show that a composition operator $C_ϕ$ is a TDZ in $\mathcal{B}(L^2(μ))$ if and only if $\frac{dμϕ^{-1}}{dμ}$ is a TDZ in $L^{\infty}(μ).$ Lastly, we determine composition operators on the Hardy spaces $\mathbb{H}^p(\mathbb{D})$ and $\ell^p$ spaces which are zero-divisors.

math.FA↗

arXiv2024

A class of Zero Divisors and Topological Divisors of Zero in some Banach algebras

In this paper, we establish necessary and sufficient conditions that must be met for weighted composition operators to act as zero divisors in $\mathcal{B}(\ell^p).$ We also give a necessary condition and a sufficient condition for a composition operators to act as zero divisors in $\mathcal{B}(L^p(μ)).$ Subsequently, we characterize TDZ in $C(X)$. Afterward, we establish that a multiplication operator $M_h$ in $\mathcal{B}(C(X))$ becomes a TDZ if and only if $h$ is a TDZ in $C(X).$ Further, motivated by the definition of TDZ, we introduce notions of polynomially TDZ and strongly TDZ and prove that every element in $C(X)$ and in $L^\infty(μ)$ is a polynomially TDZ. We then prove that a multiplication operator $M_h$ in $\mathcal{B}(C(X))$ as well as in $\mathcal{B}(L^p(μ))$ is a polynomially TDZ. Lastly, we show that each $T\in \mathcal{B}(H)$, where $H$ is a separable Hilbert space, is a strongly TDZ.

math.FA↗