When Does the Norm of the Sum Equal the Sum of Norms for Integral Operators?

[sarrah1]

Hi
I have 2 linear integral operators

$(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$
$(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$

I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$

when does $||K+M||=||K||+||M||$
thanks
sarrah

 

[Opalg]

Hi
I have 2 linear integral operators

$(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$
$(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$

I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$

when does $||K+M||=||K||+||M||$
thanks
sarrah

I believe that the answer will be "hardly ever". Notice that $\int_{a}^{b}|k(x,t)|\, dt$ and $\int_{a}^{b}|m(x,t)| \,dt$ are both functions (of $x$), and the sup norm of a sum of two functions is very seldom equal to the sum of the sup norms of the functions. In fact, equality only occurs if the two functions attain their maximum at the same point.

 

[sarrah1]

I believe that the answer will be "hardly ever". Notice that $\int_{a}^{b}|k(x,t)|\, dt$ and $\int_{a}^{b}|m(x,t)| \,dt$ are both functions (of $x$), and the sup norm of a sum of two functions is very seldom equal to the sum of the sup norms of the functions. In fact, equality only occurs if the two functions attain their maximum at the same point.

Dear Oplag
you are always of great help.
But if a,,b greater than 0 and both k(x,t) and m(x,t) are represented by series of positive terms only or negative terms only, like $k(x,t)=sinhxt$ and $m(x,t)=xt+x^3t^3/3!$ i.e. $m(x,t)$ and $k(x,t)$ reach their maximum at point b then $||M||+||K-M||=||K||. right ?

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Dear Oplag
you are always of great help.
But if a,b greater than 0 and both k(x,t) and m(x,t) are represented by series of positive terms only or negative terms only, like k(x,t)=sinhxt and m(x,t)=xt+x 3 t 3 /3! i.e. m(x,t) and k(x,t) reach their maximum at point b then $||M||+||K-M||=||K||. right ?

 

[Opalg]

But if a,b greater than 0 and both k(x,t) and m(x,t) are represented by series of positive terms only or negative terms only, like k(x,t)=sinhxt and m(x,t)=xt+x 3 t 3 /3! i.e. m(x,t) and k(x,t) reach their maximum at point b then $||M||+||K-M||=||K||. right ?

That is correct. In that case, the functions both attain their maximum at the endpoint $b$, so that is the exceptional situation when the norm of the sum is the sum of the norms.

 

[sarrah1]

Dear Oplag
final 2 small questions and I wouldn't bother you anymore:

1. When I write $||K||=max ( x\in[a,b])\int_{a}^{b} \,|k(x,t)| dt$ should I write $||K||\infty$ on the L.H.S instead of $||K||$ only

2. For 2 linear integral operators $K$ and $M$ say is $||K+M||\le||K||+||M||$ ? always
thanks
sarrah

 

[Opalg]

1. When I write $||K||=\max ( x\in[a,b])\int_{a}^{b}|k(x,t)|\, dt$ should I write $||K||_\infty$ on the L.H.S instead of $||K||$ only

Personally, I would not put a subscript $\infty$ against that norm. That subscript usually indicates a supremum norm, which in this case would mean the norm \(\displaystyle \|K\|_\infty = \max_{x,t\in[a,b]}|k(x,t)|.\) But the norm here is a "mixed norm", obtained by using the sup norm for the $x$ variable and the $L^1$-norm (the integral norm) for the $t$ variable.

2. For 2 linear integral operators $K$ and $M$ say is $||K+M||\le||K||+||M||$ ? always

It is part of the definition of a norm that it should satisfy the triangle inequality $\|A+B\| \leqslant \|A\|+\|B\|$. That certainly applies in the case of this integral norm.