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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$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not
Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $...
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Why is the numerical range of an operator convex?
Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}
It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book
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Inverse of Toeplitz Matrix Property
Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 &...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?
Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
user219526
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Eigenprojection as Contour Integral over Resolvent
Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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What is the interpretation/intuition of $e^{itA}$ for a self-adjoint unbounded operator?
Let $A : i \frac{d}{dt} : D(A) \to H^1([0,1])$ with domain $D(A)=H^1_*([0,1])=\{u \in H^1([0,1]): u(0)=u(1)\} \subseteq H^1([0,1])$. Then I know that $A$ is self-adjoint. Using the spectral theorem, ...
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Hilbert transform and Hilbert matrix
The Hilbert matrix is
\begin{bmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt]
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt]
\frac{1}{...
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Fixed point: linear operators
I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad.
Consider a space $X$ ...
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Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$
$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{S}(\mathbb{R}^k)$ denote the $k$-dimensional Schwartz space with the usual topology, and let $\mathcal{S}'(\mathbb{R}^{k}))$ denote its strong dual (...
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Eigenvector Riesz basis under operator multiplication?
I recently encountered the Riesz Spectral Operators which roughly speaking are closed operators whose eigenvectors form a Riesz basis and I became interested in when such operators can be perturbed ...
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The spectral theorem and direct integrals
I'm wondering if there are any good references that discusses the spectral theorem in terms of direct integrals? I suppose the statement would be something like this:
Let $N \in \mathcal{B}(H)$ be a ...
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When does analytic in the operator norm imply analytic in the trace class norm?
Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the ...
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Operators similar to operators with spectral radius 1
Let $A$ be a linear bounded operator acting on a Banach space $X.$ Assume the spectral radius of $A$ is equal $1.$ Do there exist invertible operators $U_n:X\to X,$ such that
$$\|U_n^{-1}AU_n\|<1+{...
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Composition of some linear differential operators $(D-A_n)...(D-A_1)$
Let $D=x\frac{d}{dx}$ and $A_i\in\mathbb{R}[[x]]$ for $i=1,...,n$. Let $B_i\in\mathbb{R}[[x]]$ for $i=1,...,n$ such that $$(D-A_n)...(D-A_1)=D^n+\sum_{i=1}^nB_iD^{n-i}.$$
Is there a well-known formula ...
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