Determine the norm of an operator Tf(t)

[Frobenius21]

Homework Statement

Hello,
I need to show that the operator
Tf(t) := g(t)f(t)

Where g is a function on
L^infinity(0,1)

-Is boundeed on the space L^2(0,1)

-and to calculate its norm.

Relevant Equations

Tf(t) = g(t)f(t)

I don't know how to start to find the bounded condition nor the norm. I thought about finding a maximal norm to show that it is bounded but I don't know how to continue.

 

[Orodruin]

What is the definition of the operator norm on $L^\infty(0,1)$? Start from there.

 

[Office_Shredder]

It sounds like f is in $L^2$ from the first part. Can you clarify what space all these objects live in?

 

[Frobenius21]

Yes, T is an operator on L^2(0,1).
I was able to show it is bounded by ||g|| in L^infinity(0,1). But I don't know how to find the norm.

 

[Office_Shredder]

Why don't you start by writing down as specifically as possible how the norm is defined for T (don't just parrot the generic definition of a norm use what you know about the space and the definition of T)

 

[Frobenius21]

I am just not able to find a function, which maximazes the operator or a sequence of functions.

Can you give me a hint if you know?

 

[Office_Shredder]

Can you write down the thing you are trying to maximize by finding a function? Once you've written down the thing you want to maximize working through it will be a lot easier.

 

[Frobenius21]

My understanding is that the norm for T has the form ||T||2 = int (|g (t) f (t)|^2) dt

Could you recommend books or resources with similar examples to the problem?

 

[Office_Shredder]

I don't think that's right. The thing on the right depends on f, the thing on the left does not. Also there's a square root involved in the $L^2$ norm that you are missing

 

[Frobenius21]

Im sorry, I missed the square root.

What do you mean by the thing on the left/right? Do you mean the left hand side and the right hand side of the equation?
Or do you mean the left and right terms in the integral?

 

[Frobenius21]

Im starting to learn the subject so I am looking for resources and books with similar examples.

 

[Orodruin]

Im sorry, I missed the square root.

What do you mean by the thing on the left/right? Do you mean the left hand side and the right hand side of the equation?
Or do you mean the left and right terms in the integral?

He means the left and right side of the equation. Your equation is not how the operator norm is defined so this is the first thing you need to find out.