Multiplication Operators

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


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
$$ {\left\| {A\sum\limits_j {{\xi _j}{e_j}} } \right\|^2} = {\left\| {\sum\limits_j {{\alpha _j}{\xi _j}{e_j}} } \right\|^2} = \sum\limits_j {{{\left| {{\alpha _j}{\xi _j}} \right|}^2}} \underline{\underline < } {\left( {\mathop {\sup }\limits_j \left| {{\alpha _j}} \right|} \right)^2} \cdot \sum\limits_j {{{\left| {{\xi _j}} \right|}^2}} = {\left( {\mathop {\sup }\limits_j \left| {{\alpha _j}} \right|} \right)^2} \cdot {\left\| {\sum\limits_j {{\xi _j}{e_j}} } \right\|^2}. $$


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Paul R. Halmos

There are no affiliations available

Personalised recommendations