|
|
|
|
Schur's condition for a matrix to be a bounded operator on 
|
(Theorem)
|
|
Theorem 0.1 Let $B$ be a matrix defined on $T\times T$ for some countable set $T$ . If there exists a positive number $C$ such that $$ \sum_{t\in T}|b(s,t)|<C \ \text{for all\ }s\ \ \ \text{and} \ \ \ \sum_{s\in T}|b(s,t)|<C \ \text{for all\ } t, $$ then $B$ is a bounded
operator on $l^2(T)$ with its operator norm $\norm{B}$ less than or equal to $C$ .
Proof. Let $x$ be a sequence in $l^2(T)$ . We have
| 2#2 |
3#3 |
|
| |
4#4 |
|
| |
5#5 |
|
| |
6#6 |
|
| |
7#7 |
|
| |
8#8 |
|
Therefore we have $\normt{Bx}\leq C\normt{x}$ for all $x \in l^2(T)$ , hence $\norm{B}\leq C$ . 1#1
|
"Schur's condition for a matrix to be a bounded operator on  " is owned by Gorkem.
|
|
(view preamble | get metadata)
| Other names: |
Schur's Lemma, Schur's Lemma for infinite matrices |
|
|
Cross-references: sequence, operator norm, bounded operator, number, positive, countable, matrix
There are 4 references to this entry.
This is version 2 of Schur's condition for a matrix to be a bounded operator on  , born on 2006-06-07, modified 2006-09-08.
Object id is 7967, canonical name is SchursConditionForAMatrixToBeABoundedOperatorOnL2.
Accessed 2110 times total.
Classification:
| AMS MSC: | 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
