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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If-and-only-if condition on the kernel for an integral operator $T:L^2 \rightarrow L^2$ to be compact

Let $\Omega \neq \emptyset$ and $\mu$ be a finite measure on $\Omega.$ We are cosidering a kernel $k$ on $\Omega,$ i.e. a symmetric, non-negative definite jointly measurable function $$k: \Omega \...
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An eigenfunction inequality for integral operators

I have an integral operator $T$ defined with respect to a positive semidefinite kernel function $k: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ probability measure $\mu(dx) = p(x) dx$ defined ...
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How to prove that the range of this integral operator $T$ is $L^2[a,b]$?

The operator is $Tf= g(t)+ \int_{a}^{t}(K(t,s)f(s)ds.$ The proof assumes that $g(t) \in L^2[a,b]$ so I believe I need to only prove that $\int_{a}^{t}(K(t,s)f(s)ds \in L^2[a,b]$. I started off this ...
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Averaged matrix exponential inverse

Let $H$ be a given symmetric matrix, i.e., $H\in\mathbb{R}^{d\times d}_{\rm sym}$, $d\geq1$. The operator $T$ is given by $$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
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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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Calculating Trace of Integral Operators

I could not figure out how the following formula can be derived: $$ \operatorname{Tr} [ W(t) Q(t) ] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d \times \mathbb{R}^d} \widehat{W}(t, q-p) \widehat{Q}(t,p,...
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Structure of the Inverse of a Fredholm integral operator of the second kind

NOTE: Cross-posted on MathOverflow I am trying to solve an equation of the form $$ (\mathbb{I} + K)\phi = f $$ where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\mathbb{R}) $ is a ...
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Compactness integral operator

The operator $Af(x)=\int_{\sqrt{x}}^{1} \frac{1}{x+y}f(y) dy, \quad f \in L^{2}([0,1])$, is bounded on $L^2([0,1])$, with $||Tf||^2=\int_{0}^{1} |\int_{\sqrt{x}}^{1} \frac{1}{x+y}f(y) dy|^2 dx \leq ......
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integral operator with continuous integrable function

Let $k: [0,\infty) \mapsto \mathbb{R}$ be continuous and $\int_{0}^{\infty} |k(x)|dx < + \infty$. Set $(Af)(x)=\int_{0}^{\infty} k(x+y) f(y) dy, f \in L^2(\mathbb{R})$. Show that $A$ is bounded and ...
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Examples of Continuous Functionals and Operators

I am currently working trough the chapter 2.10 in the book "Spectral Theory of Self-Adjoint Operators in Hilbert Space" (https://link.springer.com/book/10.1007%2F978-94-009-4586-9). This ...
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Compactness of convolution operator

Let $k$ be a bounded continuous function that is strictly positive and $\lim_{|x|\to\infty} k(x) \to 0$ and also $k \in L^1(\mathbb R)$. The question is, if the integral operator $T: C_b^0(\mathbb R) \...
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How Can One Express u(xy) as a Diagonalized Transform Kernel, K(x,y)?

Consider a projection operator $P_{u}g(x)=<g(x),u(x)>$, where $u(x)$ is an eigenfunction normalized under an inner product, $<u_{m}(x),u_{n}(x)>=\delta_{m,n}$. (ASIDE: Inner products may ...
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Is the integral operator with continuous kernel from $H^{-\frac 1 2}(\partial\Omega)$ to $H^{\frac 1 2}(\partial\Omega)$ a compact operator?

For a bounded domain $\Omega$ with $C^1$ boundary, define a integral operator $A$ with a continuous kernel $k$ satisfies $$ A f (x) = \int_{\partial\Omega} k(x,y)f(y)~{\rm d}y \quad {\rm for }~x\in\...
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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 Continuous Functional Calculus of Integral Operator

Suppose we have a symmetric bounded function $k:[0,1] \times [0,1] \rightarrow \mathbb{R}$ that induces the integral operator $T_k: L^2([0,1]) \rightarrow L^2([0,1])$, $$ (T_k f)(u) = \int_0^1 k(u,v) ...
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What is the correct formula of the kernel of the A$^*$A

If the kernel of the linear integral operator A on L$^2$(0,1) where A is a linear bounded integral operator non self adjoint? Thanks.
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A Schwartz kernel theorem for locally compact groups?

I am working through the paper A General Theory of Equivariant CNNs on Homogeneous Spaces. The paper is primarily aimed at a computer science audience. Here is the setting: we have a locally compact ...
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Showing that $||K^nf||_p \leq \frac{1}{n!}||k||_\infty^n ||f||_p$ where $K$ is the integral operator coming from the kernel $k(x,y) = \max\{0,x-y\}$

Here is what is written in my notes: $\bf{Remark.}$ There are quite a few ways to manufacture operators that contain only $0$ in the spectrum: Let $I = [0,1]$ and let $k \in \mathcal{C}(I^2)$ such ...
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Hilbert-Schmidt Integral Operator with Missing Eigenfunctions

I'm having some issues with the spectral decomposition of the integral operator \begin{equation} (Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}. \end{equation} Since \begin{equation} ...
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Pullback of integral kernel operator: how to remember the formula?

Every once in a while I have to rederive the formula for the pullback of a Euclidean integral kernel operator by a diffeomorphism. Let me focus on a simple case to make the discussion concrete. ...
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Prove the operator $\int_0^x (x-t)f(t)dt$ is well defined.

Consider the operator $T : C[0,1] \to C[0,1] $ ($C[0,1]$ has the maximum metric) defined as $$T(f)=\int_0^x (x-t)f(t)dt$$ Prove that $T$ is well defined. So essentially what I want to prove is that $...
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Integration is a compact operator on $L^p([0, 1])$

Let $p \in [1, \infty]$. I want to prove that this integral operator is compact: $$ T_p: L^p([0, 1]) \to L^p([0, 1]), \quad T(f(x)) := \int_0^x f(t)dt $$ I can prove it for $L_1$ case and I can prove ...
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Proving an expression for $T^n.$

Let $\mathcal H = L^2[0,1].$ Define an operator $T : \mathcal H \longrightarrow \mathcal H$ by $Tf(x) = \displaystyle {\int_{0}^{x} f(y)\ dy.}$ Show that for all $n \geq 1$ $$T^n f(x) = \int_{0}^{x} \...
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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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How to prove that eigenfunction of translationally invariant continuous operator $ K(t-t') $ is $ \exp(iwt) $?

I was studying a book in computational neuroscience where I came across the following equations: $$ \int{W(t,t')e(t')dt'}=\lambda e(t) $$ and read that if $ W(t,t')=K(t-t') $ then the eigenfunction $ ...
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Compute the spectrum of an explicit integral operator

Let $H = L^2[0,1]$. Define an operator $K \in B(H)$ by $Kf(x) := x^2 \cdot \int_{0}^{1} y f(y) \; \text{d} y$. Show that $K$ is compact and compute its spectrum. I already showed that $ ||K || \leq \...
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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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Mercer's theorem for zonal kernel

Let $f$ be a continuous function $[-1,1]\to\mathbb{R}$. Consider an integral operator $A$ on the unit sphere $S^{d-1}$ of $\mathbb{R}^d$, which acts on $\phi\in\mathcal{L}^2(S^d)$ as $$A\,\phi(x) = \...
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Positive definiteness wrt different borel measures

Let $K$ be a compact set in $\mathbb R^d$. Let $dx$ represent the usual Lebesgue measure and let $Q$ be a compact strictly positive definite integral operator $L^2(K,dx)\to L^2(K,dx)$: $$Q\phi(x) = \...
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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 ...
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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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1 answer
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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 $(x_n) \in D(A)$ ...
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Find spectrum of integral operator in L2

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_{-...
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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 ...
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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])...
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3 votes
3 answers
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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 ...
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2 votes
1 answer
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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 \...
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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 ...
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7 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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4 votes
1 answer
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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?
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  • 2,645
0 votes
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Operator norm of integral operator

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|...
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