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.

1,850 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
204
votes
0answers
11k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
36
votes
1answer
1k views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
23
votes
0answers
346 views

$A$ 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 $...
15
votes
0answers
1k views

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 ...
12
votes
0answers
1k views

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 &...
10
votes
0answers
274 views

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 ...
9
votes
1answer
146 views

How many connected components could the intersection of $\{A \in M_n(\mathbb R): \rho(A) < 1\}$ and an affine subspace in $M_n(\mathbb R)$ have?

Let $\mathcal E = \{A \in M_n(\mathbb R): \rho(A) < 1\}$ where $\rho(\cdot)$ is the spectral radius and $\mathcal U$ be an affine space in $M_n(\mathbb R)$. If we assume $\mathcal E \cap \mathcal U ...
9
votes
0answers
111 views

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 (...
9
votes
1answer
378 views

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 ...
9
votes
0answers
756 views

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}{...
9
votes
0answers
1k views

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 $X$ ...
8
votes
0answers
85 views

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 ...
8
votes
0answers
185 views

The sum of eigenvalues of integral operator $S(f)(x)=\int_{\mathcal{X}} k(x,y)f(y)d\mu(y)$ is given by $\int_{\mathcal{X}} k(x,x) d\mu(x)$?

Setup: Let $(\mathcal{X},d_{\mathcal{X}})$ and $(\mathcal{Y},d_{\mathcal{Y}})$ be two separable metric spaces. Let $M^1(\mathcal{X})$ be the space of Borel probability measures on $\mathcal{X}$ with ...
7
votes
1answer
247 views

If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

Let $\left(\kappa^{(n)}_t\right)_{t\ge0}$ and $(\kappa_t)_{t\ge0}$ be Markov semigroups on $(\mathbb R,\mathcal B(\mathbb R))$ for $n\in\mathbb N$ $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly ...
7
votes
0answers
117 views

Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \frac{d^2 f}{dt^2} + f$.

Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \dfrac{d^2 f}{dt^2} + f$ with $f(0) = 0$ and $f'(1) = 0$. Note: The weight ...
7
votes
0answers
276 views

If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?

Consider the following setting: Let $X, Y$ be vector spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Furthermore, let $\tau_1, \tau_2$ be locally convex topologies on $X$ and $\...
7
votes
0answers
206 views

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 ...
7
votes
1answer
576 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a densely-...
7
votes
0answers
367 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
7
votes
0answers
1k views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
7
votes
0answers
918 views

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 ...
7
votes
0answers
399 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
6
votes
1answer
73 views

On the closedness of KdV operator

Some papers that I am currently reading state that the classical KdV operator $Au=u'+u'''$ with $D(A)=\{u\in H^3(0,1)|u(0)=u(1)=u'(1)=0\}$ is a closed operator in $L^2(0,1)?$ However, no proof is ...
6
votes
1answer
189 views

Free probability: motivation for Voiculescu's free gaussian functor

Given a Hilbert space, let us denote by $\mathcal{T}(\mathcal{H})$ the full fock space $$ \mathcal{T}(\mathcal{H}) = \mathbb{C} \Omega \oplus \bigoplus_{n \geq 1} \mathcal{H}^{\otimes n}. $$ If $\xi \...
6
votes
0answers
55 views

Showing that the equation $x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i$ has a unique solution.

Exercise : Consider the infinite-dimensional system of equations : $$x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i, \quad i=1,2,3,\dots$$ We suppose that $b=(b_1,b_2,\dots) \in \ell^\infty$ and that ...
6
votes
0answers
77 views

Absolutely Continuous Spectrum of an operator $Af(x)=\sin x f(x)$

Let us consider the following operator $$ Af(x)=\sin(x)f(x), $$ in $L^2(\mathbb R)$. This operator is self-adjoint, so $\sigma(A)\subset \mathbb R.$ Solving equation $(A-\lambda I)f=g,$ we obtain $...
6
votes
0answers
97 views

given the Heat kernel corresponding to $e^{-t\Delta_q}$, find the kernel of the smoothing operator $Fe^{-t\Delta_q}$

In the proof of the Lefschetz formula for smooth maps $\phi\colon M \to M$ via the Heat Kernel (See Roe Elliptic Operators, Topology and asymptotic methods chapter 10) it's done the following ...
6
votes
0answers
163 views

Understanding the working of creation and annihilation operators on Fock space

I have some problem in understanding how exactly the creation and annihilation operators work on a Fock space. We begin with a separable Hilbert space $\mathcal H$ with inner product $\langle,\rangle$...
6
votes
1answer
124 views

Stability of point spectrum

Suppose $T$, $S$ are bounded operators on $l_2$, $a_n\to 0$ a sequence of complex numbers with the property that for any $n\in\mathbb{N}$, $T+a_nS$ has discrete spectrum and non-empty point spectrum....
6
votes
0answers
199 views

Square root of normal positive operators over real Hilbert spaces

A bounded linear operator $A$ on a Hilbert space $H$ is called a positive operator if $\langle Ax, x\rangle \geq 0$ for all $x$ in $H$. It is known that, if $A$ is a positive operator on a Hilbert ...
6
votes
0answers
207 views

Defining a trace-class operator with a Bochner integral

For complex numbers $z$ consider the system of $L^2(\mathbb{R})$-vectors with norm equal to $1$ \begin{equation} \psi_z=e^{-\frac{1}{2}|z|^2}\sum_{n=0}^{\infty}\frac{z^n}{\sqrt{n!}} |n\rangle, \end{...
6
votes
0answers
83 views

Infer $Tf=\sum_{n=1}^\infty (f,f_n) f_n$ from frame condition.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in H$ ...
6
votes
0answers
171 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space $L^2(\...
6
votes
0answers
460 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
6
votes
0answers
264 views

Simple Modules over the Weyl Algebra

Let $k$ be a field of characteristic zero and let $A_1=k\langle x,y| \, xy-yx=1 \rangle$ be the Weyl algebra. Is there a (more or less explicit) possibility of writing down all simple modules over $...
6
votes
0answers
399 views

Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if $...
6
votes
0answers
416 views

Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting

The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below. Suppose $M$ is a smooth Riemannian manifold ...
6
votes
1answer
312 views

Invertibility of Toeplitz operator in $\ell_1$

Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
5
votes
0answers
30 views

Numerical range of the first derivative operator on $\{ u \in H^1(0,1): u(1)=0 \}$

I need to calculate the numerical range of the operator $T:D(T)\subseteq L^2(0,1) \to L^2(0,1)$ defined by $$D(T):=\{ u \in H^1(0,1): u(1)=0 \}, \ Tu:=u', \ u \in D(T),$$ where $H^1(0,1)$ is the ...
5
votes
0answers
29 views

Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$

Let $Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$ with $a_j \in \mathbb{C}$. Consider the differential operators $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ and $T_\infty: D(T_\infty)\subseteq L^2[0,\...
5
votes
0answers
72 views

Proof for a classifying space for $K$ Theory.

The goal of my question is to understand a bijection between $K_0(A)$ to $[C_0(\Bbb R), M_2(M_\infty(A))]_*$ $$[C_0(\Bbb R), M_2(M_\infty(A))]_*$$ is the homotopy class of graded $*$-homomorphisms. $...
5
votes
0answers
52 views

Using abstract Hilbert spaces to solve differential equations

There are techniques for solving PDE's, such as Fock-Schwinger method in physics, which involve translating the problem from the language of distributions to the language of the abstract Hilbert ...
5
votes
0answers
77 views

Is the generator of a semigroup of bounded linear operators closed even when the semigroup is not strongly continuous?

If $E$ is a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ is a semigroup of bounded linear operators on $E$ and $(\mathcal D(A),A)$ denotes the generator of $(T(t))_{t\ge0}$, is $(\mathcal D(A),A)$ ...
5
votes
0answers
79 views

Show that no two eigenvectors of adjoint of right shift operator are orthogonal

Let $$T:\ell^2 \to \ell^2$$ is unilateral shift operator, defined by $$T(x_1,x_2,x_3......)=(0,x_1,x_2,x_3.....),$$ then show that $T$ has no eigenvalue. But every $\lambda \in \mathbb{C}$ such that $|...
5
votes
0answers
121 views

Multiplication operation of End(X) with strong topology not continuous

Let $(X, ||\ ||)$ be a normed space. $dim X = \infty$. $L(X)$ is a space of continuous (which is equialent to bounded in a normed space) linear operators with strong topology. Strong topology is ...
5
votes
0answers
102 views

trace $(ADA^{-1})=$ trace $(D)$ in infinite dimensions?

Let $X$ be a separable Hilbert space, $D$ nonnegative definite (by which I also mean self-adjoint) and trace class operator on $X$. Let $A$ be a compact and injective operator with dense range $R$, ...
5
votes
0answers
131 views

Spectrum of Scaling Operator

I'm considering this spectral problem I don't manage to solve. Suppose $T:L^2(\mathbb R)\rightarrow L^2(\mathbb R)$ is defined by: $$Tf(x)=\frac{1}{\sqrt2}f\Big(\frac{x}{2}\Big)$$ How can I calculate: ...
5
votes
0answers
78 views

What can be said about the sequence $(\left\Vert A^{n}\left( x\right) \right\Vert ^{\frac{1}{n}})_{n}$

Let $H$ be a separable infinite dimensional Hilbert space and let $A\in B(H)$ be a bounded operator. For any $x\in H$ is $(\left\Vert A^{n}\left( x\right) \right\Vert ^{\frac{1}{n}})_{n}$ a decreasing ...
5
votes
0answers
101 views

About the self-adjoint extension of one linear differential operator

I`m trying to solve some problems and I need your help. Consider $$A: \;\;-\frac{d^2y}{dx^2}: C[0,\pi] \to C[0,\pi];\;\;\; y(0)=y(\pi)=0$$. It is unbounded, closed and symmetric, but not self-...
5
votes
0answers
119 views

Is this integral operator on $L_2(0, \infty)$ compact?

Let's define $T:L_2(0,\infty) \to L_2(0,\infty)$ as $$(Tf)(x) = \int_0^\infty \frac{f(y)\sqrt{xy}}{x^2y^2+1}dy.$$ I'm interested, if this operator is compact. $T$ is integral operator with kernel $K = ...