quiksilver4210
New member
- Joined
- Dec 19, 2019
- Messages
- 1
Find norm of a linear operator
[math]A :L^2(0,1) \to L^2(0,1)[/math][math]Ax(t) = t \int\limits_{0}^{t}x(\tau)d\tau[/math]To solve it i used operator norm formula in [math]L^2 : ||A|| = max \lambda(A^*A)[/math][MATH]A^*Ax(t) =\int\limits_{t}^{1}s\int\limits_{0}^{t}tx(\tau)d\tau ds = t\int\limits_{t}^{1}\int\limits_{0}^{t}sx(\tau)d\tau ds =[/MATH](change the limits of integration) [Math]t\int\limits_{0}^{t}\int\limits_{t}^{1}sx(\tau)ds d\tau = t\int\limits_{0}^{t}x(\tau)(1-t^2/2) d\tau = t(1-t^2/2)\int\limits_{0}^{t}x(\tau)d\tau[/MATH]So i need to solve this equation and find max possible [MATH]\lambda[/MATH][MATH]t(1-t^2/2)\int\limits_{0}^{t}x(\tau)d\tau = \lambda x(t)[/MATH]But I am also not sure if previos operations was correct.
[math]A :L^2(0,1) \to L^2(0,1)[/math][math]Ax(t) = t \int\limits_{0}^{t}x(\tau)d\tau[/math]To solve it i used operator norm formula in [math]L^2 : ||A|| = max \lambda(A^*A)[/math][MATH]A^*Ax(t) =\int\limits_{t}^{1}s\int\limits_{0}^{t}tx(\tau)d\tau ds = t\int\limits_{t}^{1}\int\limits_{0}^{t}sx(\tau)d\tau ds =[/MATH](change the limits of integration) [Math]t\int\limits_{0}^{t}\int\limits_{t}^{1}sx(\tau)ds d\tau = t\int\limits_{0}^{t}x(\tau)(1-t^2/2) d\tau = t(1-t^2/2)\int\limits_{0}^{t}x(\tau)d\tau[/MATH]So i need to solve this equation and find max possible [MATH]\lambda[/MATH][MATH]t(1-t^2/2)\int\limits_{0}^{t}x(\tau)d\tau = \lambda x(t)[/MATH]But I am also not sure if previos operations was correct.
