On C[0, 1] define the operators S and T bySf(x) = f(t)dt. Tf(x) = af (1), a € [0, 1].r(c) calculate norms of S, T, ST and TS.by

Given two operators S,T on the set of continuous functions .Consider the norm as .Also given To…

Answered: On C[0, 1] define the operators S and T…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

On C[0, 1] define the operators S and T by
Sf(x) = f(t)dt. Tf(x) = af (1), a € [0, 1].
r
(c) calculate norms of S, T, ST and TS.
by

Expert Solution

Step 1: Definition of the norm and norm of an operator

Given two operators S,T on the set of continuous functions $C open square brackets 0 comma 1 close square brackets$ .

Consider the norm as $open double vertical bar f open parentheses x close parentheses close double vertical bar equals integral subscript 0 superscript 1 open vertical bar f open parentheses x close parentheses close vertical bar d x$ .

Also given $S f open parentheses x close parentheses equals x integral subscript 0 superscript 1 f open parentheses t close parentheses d t comma T f open parentheses x close parentheses equals x f open parentheses x close parentheses comma space space space space space space space space space space space x element of open square brackets 0 comma 1 close square brackets$

To find the norms of $S comma space T comma space S T comma space T S$ .

If $T$ is a transformation on a normed space $V$ , then the norm of the transformation is defined as

$table row cell open double vertical bar T close double vertical bar end cell equals cell s u p subscript open double vertical bar v close double vertical bar equals 1 end subscript open double vertical bar T open parentheses v close parentheses close double vertical bar end cell end table$ where $v element of V$ .