Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

learn more… | top users | synonyms

0
votes
0answers
37 views

Non Normal operator [on hold]

Standard example for non normal operator is the shift operator. It is continous but the image of the left shift is not dense. Can we have an example of a non normal operator $A$ which is continuous ...
3
votes
1answer
64 views

Composition of spectral measures

Let $f: \mathbb{R}\rightarrow \mathbb{C}$ be a measurable function, $H$ some Hilbert space and $$ f_E := \int_{\mathbb{R}} f dE$$ for some spectral measure $E$. Now, my question is: When do we have ...
0
votes
0answers
95 views

Anticommuting operators with positive properties

Which classes of $M\in \mathsf M_k(\Bbb R)^{n\times n}$ admit solutions $N\in \mathsf M_k(\Bbb R)^{n\times n}$ such that $$(M\otimes N+N\otimes M)(u\otimes u)=0$$ forall $u\in \mathsf D_k(\Bbb ...
1
vote
0answers
90 views

A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map ...
2
votes
1answer
145 views

Asymptotic behaviour of eigenvalues

If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
0
votes
0answers
17 views

Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
0
votes
0answers
48 views

Hilbert scales of covariance operators

Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that ...
1
vote
0answers
70 views

Normal points of an operator and discrete eigenvalues

Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively. As a graduate student entering the field of ...
1
vote
1answer
99 views

Approximation Property: Decomposition

This thread originated from MSE: Approximation Property: Decomposition Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional ...
0
votes
0answers
66 views

Approximation Property: Characterization

Problem Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$. Suppose it has the approximation ...
0
votes
0answers
67 views

Compact Approximation

This thread originated from MSE: Compact Approximation This is meant as lemma for: Approximation Property Given a Banach space $E$. Denote compact domains by $\mathcal{C}$. Denote compact ...
4
votes
0answers
80 views

Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
1
vote
0answers
60 views

Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following: $$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$ Consider the one dimensional case. Then the free ...
0
votes
1answer
146 views

Totally non hereditary $C^{*}$-subalgebras

Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
0
votes
0answers
36 views

Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
0
votes
0answers
98 views

Perturbation of Laplacian via Kato-Rellich theorem

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
3
votes
1answer
178 views

adjoint of this closed (?) operator

I am currently dealing with an unbounded operator $T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow ...
1
vote
0answers
35 views

Recent Survey on Dynamics of Linear Operator

I'm studying Linear Dynamics using the textbook Linear Chaos by grosse erdmann. I'm looking for a recent encyclopaedic article/survey which gives me a big picture of the area. It seems erdmann and ...
2
votes
1answer
98 views

Proper domain for operators

in this paper on arxiv in equation 27, two operators $$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$ and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...
0
votes
0answers
89 views

Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...
5
votes
2answers
164 views

Two-sided ideals of $B(H)$ are hereditary

I seem to recall that (not necessarily closed) two-sided ideals of $B(H)$ are hereditary. Is that true? If it is, can anyone post a proof/reference?
1
vote
3answers
289 views

Decompose the Laplacian

Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator $A$ and its adjoint $A^*$? I am interested in finding such a decomposition, but I could not get one by ...
0
votes
0answers
50 views

Analytic functions space on Riemann surface

I have some questions about the analytic function space on Riemann surface and distinguished varieties: Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...
1
vote
2answers
252 views

Witten index non-trivial in the context of Quantum Mechanics?

Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$. I will now consider the one-dimensional case on a compact set: So ...
4
votes
2answers
191 views

The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...
7
votes
1answer
380 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
0
votes
0answers
46 views

System of integral equations

Let $K_1,K_2,K_3,K_4$ be integral operators. I'm interested in the following system of integral equations. $$\begin{cases} g_1 = K_1f_1 + K_2f_2 \\ g_2 = K_3f_1 + K_4f_2 \end{cases}$$ I'm ...
15
votes
2answers
1k views

The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is ...
5
votes
0answers
161 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
6
votes
1answer
193 views

Domains of raising and lowering operators in QM

I should add that what is known in physics as SUSY QM, is often called the Crum-Darboux method in Mathematics, so don't be confused about this. Let $H : \operatorname{dom}(H) \subset L^2(\Omega) ...
0
votes
1answer
57 views

Galerkin Projection on Integral Operators

I am looking at a research paper that mentions integral operators (which in this case is brought up in reference to shading equations that are integral operators) and it says that we can create a ...
2
votes
1answer
132 views

Eigenfunctions of an infinite summation operator

I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*: $f \rightarrow \sum_{1}^{\infty} f(nx)$ So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$ ...
2
votes
2answers
279 views

Geometrical interpretation of a Schrödinger operator

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
2
votes
1answer
113 views

Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra

As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...
3
votes
1answer
117 views

Simple $Z^{*}$ algebra

What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
2
votes
0answers
70 views

A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...
3
votes
1answer
158 views

A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$. To what extent ...
4
votes
0answers
106 views

The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number. Now ...
1
vote
0answers
137 views

Applications of composition operators on Sobolev spaces

I wold like to know some examples where composition operators on Sobolev spaces are useful. I'm in the following situation. $L^1_p(D)$ - homogeneous Sobolev space, in other words space of locally ...
2
votes
1answer
89 views

Limit-circle and limit-point at endpoints

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
3
votes
2answers
167 views

Schrödinger operators on a sphere

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
2
votes
1answer
74 views

A Possible characterization of F.D or AF commutative $C^{*}$ algebras

By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra. Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...
0
votes
0answers
115 views

Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$ $$f(\alpha x+(1-\alpha)y)\leq ...
6
votes
1answer
205 views

Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
0
votes
0answers
42 views

Self-adjointness of the components of the magnetic derivative

On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...
11
votes
1answer
366 views

What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
2
votes
0answers
83 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform ...
1
vote
0answers
69 views

Semigroups on Banach Lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that $$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$ Where $X_+$ denotes the positive ...
1
vote
0answers
87 views

Norm of the operator acting on spinor bundle

Please forgive me if the question is too elementary, but however I was unable to manage by myself. The question comes from J.Varilly, H.Figueroa and J. Gracia-Bondia book "Elements of noncommutative ...
6
votes
0answers
81 views

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...