**0**

votes

**0**answers

20 views

### Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions.
For all $f\in L^2(\Omega)$, we denote by $S(t)f$ the solution of the equation
$$
...

**4**

votes

**1**answer

136 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

170 views

### Is the residual spectrum of every power bounded operator contained in the open unit disk?

$\newcommand{\cH}{\mathcal{H}}
\newcommand{\CC}{\mathbb{C}}$
Let $\cH$ be a Hilbert space. A linear operator $T: \cH \to \cH$ is said to be power bounded if $\sup_{n \geq 0} \|T^n\| < \infty$.
If ...

**-3**

votes

**1**answer

157 views

### Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb ...

**2**

votes

**2**answers

191 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 ...

**2**

votes

**1**answer

68 views

### Domain of fractional powers of operators

Let $A$ and $B$ be non-negative ($(A x, x) \geq 0$ for all $x \in \mathcal{D}(A)$, similarly for $B$) densely defined self-adjoint operators on a Hilbert space $H$. Then the spectral theorem defines ...

**1**

vote

**0**answers

22 views

### Equivalence of fractional power of second-order positive differential operator as pseudodifferential operator and a fractional definition

Let $A$ be a second-order differential operator on a closed manifold $M$ satisfying
$$(Au,u) \geq 0$$
with $A=-\Delta$ the Laplace-Beltrami the model case. One can define for $s \in (0,1)$ the ...

**4**

votes

**1**answer

112 views

### Operator on a Banach space

Let $T$ be a continuous operator on a Banach space $V$. Assume there exist $T$-stable finite-dimensional subspaces $V_i$ such that $\bigoplus_{i=1}^\infty V_i$ is dense in $V$, on $V_i$ the operator ...

**2**

votes

**1**answer

146 views

### Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...

**7**

votes

**0**answers

188 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?(For ...

**0**

votes

**0**answers

13 views

### Unboundedness of Laplacian [migrated]

I am currently considering the following operator ("modified Laplacian"):
$T \colon \left( W^{2,2}(\mathbb{R}), \| \cdot \|_{L^2} \right) \longrightarrow \left( L^2(\mathbb{R}), \| \cdot \|_{L^2} ...

**0**

votes

**1**answer

257 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 ...

**1**

vote

**1**answer

78 views

### Element Analytic, C*-dynamical system

good night...
I was looking into the Pedersen Book, $C^{*}$-Algebras and their automorphism
groups, and found the definition of analytic elements $x\in A$, where $(A,\alpha)$ is a $C^{*}-$dynamical ...

**10**

votes

**3**answers

2k views

### When is an integral transfrom trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...

**3**

votes

**1**answer

155 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**0**

votes

**1**answer

118 views

### Schatten $p$-classes for small $p$

Suppose $\mathcal H$ is a separable Hilbert space and $T$ is a compact self-adjoint operator on $\mathcal H$. Let $\{e_n\}$ be an orthonormal basis for $\mathcal H$.
Fix $1<p<2$.
Does ...

**0**

votes

**0**answers

27 views

### On the induced norms of stochastic operator and its adjoint operator

The background: when studying the paper published in Automatica named '$H_{\infty}$ control and filtering of discrete-time stochastic systems with multiplicative noise' (volume 37, pp. 409-417), I ...

**0**

votes

**1**answer

76 views

### Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...

**0**

votes

**0**answers

73 views

### Alternating projections and trace preserving maps

Let $\{\Pi_i\}_{i=1}^N$, $\Pi_i\in \mathbb{C}^{n\times n}$ be a set of orthogonal projections. By the Von Neumann alternating projection theorem, it holds that
$$
...

**6**

votes

**1**answer

279 views

### An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...

**2**

votes

**1**answer

155 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 $e^{tA}$ ...

**2**

votes

**1**answer

79 views

### Integral kernels of self-adjoint operators [closed]

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...

**4**

votes

**5**answers

466 views

### About the trace class operators and their motivation

What is the motivation for trace class operators? Can any body suggest the most general and standard reference that includes Schatten p class operators as well.
I have following references
...

**1**

vote

**0**answers

85 views

### What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator :
...

**0**

votes

**0**answers

121 views

### Heat asymptotics

Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...

**1**

vote

**1**answer

94 views

### On the equality Tr(Af) = Tr(fA)

Consider the Hilbert space $H = L^2(\mathbb{R})$, and a bounded operator $A \in B(H)$ which satisfies:
$$
\forall f \in H, \quad Af \text{ is trace class and } Tr(Af) < C \| f \|_{H},
$$
where $f$ ...

**0**

votes

**0**answers

55 views

### On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$.
Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively.
Let ...

**0**

votes

**0**answers

86 views

### Boundedness of heat semigroup on $L^1(\Omega)$

On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...

**1**

vote

**0**answers

29 views

### Simplifying closed form for Meta Operator

I was consider the set of linear operators:
$$O_{a,k} = \frac{f(ax^k) - f(x)}{ax^k - x} $$'
Particularly I am looking for the closed forms of the eigenfunctions of this operator, that is the ...

**3**

votes

**1**answer

98 views

### Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces.
Let $B_1, \ldots, B_k \in B(\mathcal{H}).$
Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via ...

**4**

votes

**2**answers

617 views

### Sum of two closed operators closable

I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:
Is the sum of two closed operators closable? If not, give an example of two ...

**3**

votes

**1**answer

97 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 ...

**8**

votes

**2**answers

580 views

### Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate ...

**1**

vote

**2**answers

143 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

79 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

135 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 ...

**4**

votes

**0**answers

77 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, ...

**12**

votes

**1**answer

542 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

186 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

**1**answer

173 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

27 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

110 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 ...

**2**

votes

**1**answer

260 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 ...

**2**

votes

**1**answer

136 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 := ...

**0**

votes

**1**answer

142 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 ...

**3**

votes

**0**answers

91 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 ...

**5**

votes

**1**answer

262 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

160 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

105 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 ...

**3**

votes

**1**answer

79 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 ...