# Multiplication Operators

• Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 19)

## Abstract

If A is a diagonal operator, with Ae j = α j e j , then
$$\left| {{\alpha _j}} \right| = \left\| {{\alpha _j}{e_j}} \right\| = \left\| {A{e_j}} \right\|\underline{\underline < } \left\| A \right\| \cdot \left\| {{e_j}} \right\| = \left\| A \right\|,$$
So that {α j } is bounded and sup j |α j | ≦ ║A║. That reverse inequality follows from the relations
$${\left\| {A\sum\limits_j {{\xi _j}{e_j}} } \right\|^2} = {\left\| {\sum\limits_j {{\alpha _j}{\xi _j}{e_j}} } \right\|^2} = \sum\limits_j {{{\left| {{\alpha _j}{\xi _j}} \right|}^2}} \underline{\underline < } {\left( {\mathop {\sup }\limits_j \left| {{\alpha _j}} \right|} \right)^2} \cdot \sum\limits_j {{{\left| {{\xi _j}} \right|}^2}} = {\left( {\mathop {\sup }\limits_j \left| {{\alpha _j}} \right|} \right)^2} \cdot {\left\| {\sum\limits_j {{\xi _j}{e_j}} } \right\|^2}.$$

## Keywords

Multiplication Operator Positive Measure Cauchy Sequence Measurable Subset Reverse Inequality
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.