Task 8 (Translation semigroup) Let BUC(R) BUC'(R) {f € C(R); ƒ is bounded and uniformly continuous} {f € C* (R); ƒ and f' are bounded and uniformly continuous} equipped with the norms || f|| BUC(R) := sup |f(x)|, xER ||f||BUC (R) := sup(|f(x)|+ f'(x)|) xER (these are Banach spaces). Let (T(t)f)(x) := f(x+t), t > 0, ƒ € BUC(R). Show that T is Co-semigroup on BUC(R) generated by the classical derivative operator ð : BUC'(R) → BUC (IR). Show that T is not strongly continuous on Loo (R).
Task 8 (Translation semigroup) Let BUC(R) BUC'(R) {f € C(R); ƒ is bounded and uniformly continuous} {f € C* (R); ƒ and f' are bounded and uniformly continuous} equipped with the norms || f|| BUC(R) := sup |f(x)|, xER ||f||BUC (R) := sup(|f(x)|+ f'(x)|) xER (these are Banach spaces). Let (T(t)f)(x) := f(x+t), t > 0, ƒ € BUC(R). Show that T is Co-semigroup on BUC(R) generated by the classical derivative operator ð : BUC'(R) → BUC (IR). Show that T is not strongly continuous on Loo (R).
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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Transcribed Image Text:Task 8 (Translation semigroup)
Let
{f € C(R); ƒ is bounded and uniformly continuous}
{f € C' (IR); ƒ and f' are bounded and uniformly continuous}
BUC(R)
:=
BUC'(R) :=
equipped with the norms
||f | BUC(R) := sup |f (x)|,
||f |BUC (R) := sup(|f(x)|+ f'(x)|)
xER
xER
(these are Banach spaces).
Let (T(t)f)(x) := f(x+t), t > 0, f e BUC(R). Show that T is Co-semigroup on BUC(R) generated
by the classical derivative operator ð : BUC (IR) → BUC(R). Show that T is not strongly continuous
on Loo (R).
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