**3**

votes

**0**answers

48 views

### Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

**1**

vote

**1**answer

101 views

### compact inclusion of domains of unbounded operators

Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold.
Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset ...

**3**

votes

**1**answer

79 views

### Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...

**0**

votes

**0**answers

18 views

### Weak continuous convergence of operators [migrated]

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$.
Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$.
Let $x_n \rightharpoonup x$ in ...

**2**

votes

**1**answer

115 views

### Contraction semigroup

Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator which generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If additionally, we have that $e^{tA}$ ...

**0**

votes

**0**answers

67 views

### First order elliptic pseudodifferential operator and Sobolev space [closed]

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...

**1**

vote

**2**answers

119 views

### Sum of two unbounded self-adjoint operators

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are ...

**2**

votes

**1**answer

60 views

### Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.
And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...

**4**

votes

**1**answer

114 views

### Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...

**3**

votes

**0**answers

67 views

### Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...

**0**

votes

**1**answer

150 views

### Operator theory of the Hessian

How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...

**12**

votes

**1**answer

483 views

### Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”?

This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0.
To be ...

**3**

votes

**1**answer

144 views

### Sobolev spaces on compact manifolds

Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent ...

**1**

vote

**0**answers

24 views

### Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...

**3**

votes

**1**answer

67 views

### Domain of square root of a self-adjoint positive operator [closed]

Let $A \geq 0$ be a densely defined self-adjoint positive operator on a Hilbert space $H$ obtained by Friedrichs extension, and let $Q$ be the densely defined quadratic form associated to $A$, that ...

**0**

votes

**0**answers

25 views

### Fractional Laplacian on the torus [migrated]

Consider the Laplacian on the $n$ dimensional torus $T$, given by $-\Delta : L^2 \rightarrow L^2$. Let the domain of $-\Delta$ be all $C^\infty$ functions initially. Now consider the Friedrichs ...

**2**

votes

**1**answer

216 views

### Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...

**3**

votes

**1**answer

171 views

### A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$
Does for any ...

**3**

votes

**0**answers

77 views

### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a ...

**2**

votes

**1**answer

133 views

### Non-closability of an operator

Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by
$$T(u_1,u_2) := (u_1' + au_2',0)$$
on $\textrm{Dom} \,T := ...

**3**

votes

**1**answer

74 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

**0**answers

101 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

**0**answers

106 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

**1**answer

159 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

**0**answers

34 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

**0**answers

53 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

**0**answers

87 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

**1**answer

107 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

**0**answers

72 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

**0**answers

69 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

**0**answers

89 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 $S_t u(\xi)$, ...

**5**

votes

**0**answers

141 views

### Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements are path connected.
Is the tensor product of two path connected algebra, a path connected algebra?
What ...

**1**

vote

**0**answers

86 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

**1**answer

153 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

**0**answers

43 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 ...

**1**

vote

**0**answers

111 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

**1**answer

186 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

**0**answers

44 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

**1**answer

105 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

**1**answer

138 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

**2**answers

177 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

**3**answers

293 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

**0**answers

56 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

**2**answers

258 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

**2**answers

206 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

**1**answer

417 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 ...

**15**

votes

**2**answers

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

**0**answers

170 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 ...

**5**

votes

**1**answer

254 views

### Domains of raising and lowering operators in QM

Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and ...

**0**

votes

**1**answer

64 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 ...