Multiplication and Composition Operators on Weak $L_p$ Spaces

Abstract

In a self-contained presentation, we discuss the $\hbox {Weak}\,L_p$ spaces. Invertible and compact multiplication operators on $\hbox {Weak}\,L_p$ are characterized. Boundedness of the composition operator on $\hbox {Weak}\,L_p$ is also characterized.

Appendix

Definition 7.1

Let $T:X\rightarrow X$ be an operator, a subspace $V$ of $X$ is said to be invariant under $T$ (or simply $T-$invariant) whenever

$$\begin{aligned} T(V)\subseteq V. \end{aligned}$$

Theorem 7.2

Let $T:X\rightarrow X$ be an operator. If $T$ is compact and $M$ is a closed $T$-invariant space of $X$. Then $T\Big |_M$ is compact.

Proof

Let $\{x_n\}_{n\in \mathbb {N}}$ be a sequence in $M\subseteq X$. Then $\{x_n\}_{n\in \mathbb {N}}\subseteq X$, thus there exists a subsequence $\{x_{n_k}\}_{k\in \mathbb {N}}$ of $\{x_n\}_{n\in \mathbb {N}}$ such that $T(x_{n_k})$ converges in $X$ but $T(x_{n_k})\subseteq T(M)$, since $\{x_{n_k}\}\subseteq M$. Then $T(x_{n_k})$ converge on $\overline{T(M)} \subseteq \overline{M}=M$.

Therefore $T(x_{n_k})$ converge on $M$, hence $T\Big |_M$ is compact. $\square$