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1answer
36 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 ...
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0answers
29 views

Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} ...
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1answer
33 views

Prove an integral operator is compact

The statement is like this, $K\subset\mathbb{R}$ is compact, the operator $A:L^\infty(K)\mapsto L^\infty(K)$ is defined by $f(x)\mapsto\int_K k(x,y)f(y)dy$. For $x\neq y$, $|k(x,y)|\leq ...
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1answer
64 views

Does a compact operator always have a kernel?

I am sorry if this question is stupid..... I raise it when I read Lax's book Functional Analysis. We know that some integral operators are compact, for example an integral operator from $L^2[Y]$ to ...
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2answers
89 views

What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
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0answers
38 views

Norm of an integral operator

I have an exercise that I need to solve and I can't finish it. Let $k \in \mathcal{C}([0,1] \to \mathbb{R})$. Proove that this operator : $$ \begin{array}{ccccc} T & : & ...
2
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1answer
55 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
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0answers
24 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
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0answers
35 views

Kernel of Integral operator

Let $H: L^2(M) \longrightarrow L^2(M)$ be a bounded operator. Here, $M$ can be a Riemanniannian manifold, or some open subset of $\mathbb{R}^n$. Question: What can I say about the Schwartz Kernel $k$ ...
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1answer
21 views

Bounded operator on continuous functions

Let $X=C([0,1])$ and $T: X \rightarrow X$ defined as $$(Tf)(t)=f(t)+f(0)$$ Prove $T$ is bounded. I was thinking about using the fundamental theorem of calculus in order to get some bounds on $f(0)$ ...
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1answer
70 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 ...
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0answers
20 views

Computing the asymptotic spectrum of a negative distance kernel

Consider the following integral operator: $$K(f) : x \mapsto\int_0^1 K(x,x')f(x') dx', \quad \text{where} \quad K(x,x') = - |x-x'|^{3/2}.$$ The kernel is sometimes referred to as a negative ...
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0answers
27 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
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0answers
34 views

Regulairty of eigenfunctions of singular integral equations

Can you provide a proof or a reference, to study from, for the following problem: Let $\Gamma$ be a real analytic rectifiable closed curve in the plane, $ds$ is the arc-length , and kernel $K(z,w)$ ...
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2answers
206 views

Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$ T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right. $$ where ...
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1answer
71 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
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0answers
22 views

Is that operator positive-definite?

Let's consider the integral operator $\phi(x) = \int\limits^1_0\psi(y)\ln\Bigl(\Bigl|\frac{\sqrt{1-x^2}+\sqrt{1-y^2}}{\sqrt{1-x^2}-\sqrt{1-y^2}}\Bigr|\Bigr)\,dy$. How to check is this operator ...
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1answer
224 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 ...