# Questions tagged [integral-operators]

This tag is for questions relating to integral operators, which are an important special class of linear operators that act on function spaces.

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### Eigen functions of a non self-adjoint integral operator.

Let us define the operator from the set of smooth complex valued functions from $[0,1]$ to itself as $$L[f](u) = \int_{0}^{1} e^{-j2\pi(y-x)^2} f(x)dx, \quad 0\leq u \leq 1 .$$ Here $j=\sqrt{-1}$. I ...
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### Asymptotics of singular values ​of the operator

In space $L^2(B_R(0))$ consider operator $Tx(t)=\displaystyle\phi(t)\int\limits_{B_R(0)}\cos(|t|^{1/2}|s|^{1/6})\psi(s)x(s)ds$, where $\phi, \psi \in L^2$. It is clear that this operator is compact. ...
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### Kernel operators on AL-spaces

Let $E$ be an AL-space. For simplicity $E=L_{1}(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a strictly localizable measure space. Let $T:E\rightarrow E$ be a bounded kernel operator on $E$ with ...
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### Compact integral operator on $H^1(\mathbb{R})$

Consider the operator $${\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy.$$ Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself? To give some context to the problem ...
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1 vote
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### Numerical methods for computing singular value decomposition of integral operator

Consider a Hilbert-Schmidt integral operator $T:L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$, whose explicit mapping is given by \begin{align} f(y)\mapsto \int k(x,y) f(y) dy. \end{align} Hopefully,...
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### Lower-bound for $\inf_{f \in H} \|\nabla f\|_{L^2(\tau_d)}/\|f\|_H$ for an RKHS $H$ induced by dot-product kernel

Let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $\tau_d$ be the uniform probability measure thereupon. Let $H_K \subseteq L^2(\tau_d)$ be an RKHS of square $\tau_d$-integrable functions, ...
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### Kernel of a linear functional

I have $L^2((0,1))$ space and functional $$F(f) = \int_0^1 f(x)(1-x) dx.$$ How do I find the kernel? From definition we want to find $f$ such that $\int_0^1 f(x)(1-x) dx = 0$. I suspect we want the ...
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### Dirac $\delta$-function as a limit of matrices, and eigenvalues of integral kernels.

I am interested in calculating the eigenvalues of integral kernels, but I figured it would be useful to get some intuition about the simplest one first: the identity operator. The interval $[-L,L]$ ...
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### Interchange Integration and Minimization

I am new to calculus of variations and I have a problem of the type \begin{equation*} \text{min}_{g\in L^2(\mathbb{R})} \int_{\mathbb{R}} f(x,g(x))\, \mathrm dx, \end{equation*} where $f$ is ...
1 vote
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### Mercer theorem and uniformly bounded continuous function

Consider a continuous symmetric real function $k$ on $I\times I$, where $I$ is a compact real interval. Let $K$ be the integral operator whose kernel is $k$. Assume that $K$ is strictly positive ...
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### Example of a "weak" Compact operator

Let $\mathcal{H}$ be a separable Hilbert space with ONB $\{e_n\}_n$. Following my previous question, I would like to find an example of a bounded operator $A$ such that $A$ is compact. $A$ is not $p$-...
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### Extension of the Schur product theorem to operators

Given two $n\times n$ matrices $A$ and $B$, define their Hadamard product $A\circ B$ as the element-wise product, i.e. $$(A\circ B)_{ij} = A_{ij}B_{ij}\,.$$ A well known result is the Schur product ...
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### Conditions for compactness of operator

Let $A$ be an bounded operator on a Hilbert space with ONB $\{e_n\}_n$. I am looking for precise conditions on $\langle e_n, A e_m \rangle$ to guarantee that $A$ is compact (i.e. the limit of finite ...
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### Integral calculation for eigenvalues

I am trying to solve the following integral $\int_{0}^{2\pi}{\left|\int_{0}^{2\pi}{e^{ik\cos\left(\theta-\varphi\right)}\cos\left(n\theta\right)}d\theta\right|^2 d\varphi},\ \ n=0,1,2,.......$ which ...
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### Integral operator is not closable

Let $A$ be a linear operator from $X$ to $Y$ with domain $D(A)$. I've learned the following characterization for closable operators: $A$ is closable if and only if for every sequence $(x_n) \in D(A)$ ...
Please help me find the spectrum of operator $A$, I do not speak English well. In Russian forums I did not find the answer. Find spectrum of integral operator in $L_2$($R$): $$(Ax)(t) = \int\limits_{-... 0 votes 1 answer 107 views ### Prove that a specific non self-adjoint linear operator has real eigenvalues I am dealing with a Fredholm operator A, whose kernel K is non symmetrical. However when I discretise A and compute the eigenvalues of the non symmetric matrix I obtain, I observe that A only has ... 0 votes 1 answer 87 views ### Contraction mapping in C([0,1]) Suppose T is an operator on C([0,1]) defined by (Tu)(t) = \displaystyle\int_{0}^{t} u(x)^2\,\mathrm dx. Show that T is a contraction mapping on the closed ball of radius \dfrac14 in C([0,1])... 3 votes 3 answers 876 views ### How to Find the Spectrum of an Integral Operator I need to find the spectrum of an operator T: C([0,1]) \to C([0,1]) defined by (Tf)(t) = \int_0^t f(x) dx. I know that the spectrum is the set of all values \lambda such that \lambda I - T is ... 2 votes 1 answer 63 views ### Are Hilbert Scmidt integral operators on separable compact Hausdorff spaces in the Hilbert Schmidt class? Let X be compact separable Hausdorff space with a positive Borel measure \mu. Assume L^2(X) is separable. Consider a function K: X \times X \to \mathbb{C} with K \in L^2(X \times X , \mu \... 1 vote 2 answers 399 views ### Integral operator has no eigenvalue Let V be the vector space of all real valued continuous functions. Prove that the linear operator \displaystyle\int_{0}^{x}f(t)dt has no eigenvalues. This question is same as Prove that the ... 7 votes 1 answer 205 views ### An Orthogonality Problem of Eigenfunctions of homogeneous Fredholm equation Suppose we have a integral equation$$\int_{-1}^1 \frac{\text{sin }c(x-y)}{\pi (x-y)}\psi(y)dy=\lambda \psi(x),\quad|x|\le1.$$By the Fredholm equation theory, we know that this equation has ... 1 vote 1 answer 250 views ### Eigenvalues and eigenfunctions of an integral operator Let T be an integral operator with kernel K(x,y)=e^{|x-y|} on L^2(-1,1). How can we find the eigenfunctions and eigenvalues of T? Even though I am not sure whether the following arguments are ... 4 votes 1 answer 139 views ### Point spectrum of an integral operator Let we have$$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
Suppose we have $X=L^2([0,1];\mathbb{R})$ and \begin{equation} T:X\rightarrow X, \ Tf(x)=\int_0^1x^2yf(y)dy. \end{equation} Show that $T$ is compact and determine $||T||.$ I already have that \$||T|...