# Questions tagged [operator-theory]

Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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### Is it necessary to use Arzela Ascoli to show there is convergent subsequence of $(Tf_{n})_{n}$

Let $(f_{n})_{n}\subseteq C([0,1])$ be an arbitrary sequence such that $\vert \vert f_{n} \vert \vert_{\infty}\leq 1$ for all $n \in \mathbb N$ and $k \in C([0,1]^{2})$. Define the bounded linear ...
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### How to interpret composition of Integral operators

Let $Tf(y):=\int_A k_T(x,y)f(x)dx$ and similarly for $S$ with kernel $k_S$. How do I interpret $STf(y)$? The book I'm reading states that $k_{ST}(x,z):=\int k_S(y,z)k_T(x,y)dy$. Why is that?
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### Set of tensor products

Let $A,B$ be operators that come with an argument from $L^2(\mathbb{R})$. Then we consider the set $$\mathcal{A} = \{ e^{iA(f)}\otimes e^{iB(g)}, \ f,g\in L^2(\mathbb{R})\}$$ and in particular the ...
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### Orthogonal function expansions and eigenvalues

Consider expanding some function $f: [-1,1] \rightarrow \mathbb{R}$ in terms of say Legendre functions: $$f(x) = \sum_{n=0}^{N} a_n P_n(x)$$ where the truncation limit $N$ is some large number ...
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### Checking injectivity of linear operator under assumptions on the adjoint

Motivation (skip if you want). I am reading a classic paper by Donaldson about orientation of Yang-Mills moduli space https://projecteuclid.org/euclid.jdg/1214441485 I am having an hard time in ...
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### Surjectivity of Compact Operators on an Infinite Dimensional Vector Space

A pretty standard result says that if $T$ is a compact operator acting on a Banach space, then $T$ cannot be surjective. The proofs I've seen use the open mapping theorem and the fact that a ...
Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact operator: (i) The operator $K$ is called relatively compact with respect to $T$, ...
### Determining $\| \delta_{n} \|_{*}=\frac{n}{2}$
Let $X:=C([-1,1])$ and equip it with $\| \cdot \|_{1}$. Further, let $\delta_{n}: X \to \mathbb C$ be a linear functional, such that for $n \in \mathbb N$, \$\delta_{n}(f)=\frac{n}{2}\int_{-\frac{1}{n}}...