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Questions tagged [integral-operators]

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7
votes
0answers
616 views

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
2
votes
1answer
268 views

Does an integral operator with a symmetric integrable kernel have to be bounded on $L^2$?

Suppose $K(x,y)$ is a symmetric kernel. Let $\phi\in L^2(\Omega)$, where $\Omega$ everywhere is a domain in $R^n$. Can $\int_{\Omega}K(x,y)\,\phi(y)\,dy$ belong to $L^2$? In other words can an ...
6
votes
2answers
182 views

How can I show that this operator is bounded on $L^2$?

Consider the integral operator $$Tf(x) = {\int}_{-\infty}^{\infty} \frac{\sin(x - y)}{x - y}f(y)dy$$ How can I show that $T$ is bounded on $L^2$? I know that bounded means there is a ...
13
votes
1answer
448 views

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators $$...
4
votes
2answers
442 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
2
votes
0answers
99 views

Let $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)\,\mathrm{d}t}$ (the Hardy operator) find the norm of $T$ on $L^p$ [duplicate]

We have the operator $T: L^p(\mathbb{R}^+) \to L^p(\mathbb{R}^+) $ with $p \in (1,+\infty)$, defined by $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)dt}$. We define $\tilde{f}(x)=e^{x/p}f(e^x)$ for all $f \in ...
0
votes
1answer
363 views

Norm of Integral Operator on $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$

There are similar question but the characterization of the space $E$ that I have gives me problem in computing the actual norm. Let $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$ with the usual $\parallel \cdot\...
4
votes
1answer
1k views

Adjoint of an integral operator

I'm reading through a text about integral operators and I've come across the following theorem: Let $k:\mathbb{R}^2\rightarrow\mathbb{C}$ be a kernel, $T:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ ...