**3**

votes

**0**answers

144 views

### An equivalent relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**1**

vote

**0**answers

75 views

### Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$.
Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...

**2**

votes

**0**answers

30 views

### Restriction of a semigroup to a form domain

Say, we have a Hilbert space $H$ with a semibounded self-adjoint
operator $A:D(A)\to H$ generating a strongly continuous semigroup
$T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of ...

**0**

votes

**0**answers

44 views

### Generating a series representation for the inverse of the operator $f(f)$

I was considering the following problem:
Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...

**1**

vote

**1**answer

67 views

### Sum of two surjective operators

It is well-known that the sum of two surjective operators isn't (in general) a surjective operator (for example consider $A+(-A)$). When it happens that the sum of two surjective operators is still ...

**0**

votes

**0**answers

23 views

### Algorithm for Finding Eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like,
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in ...

**1**

vote

**0**answers

26 views

### Behavior of fundamental solution for parabolic equation on non compact complete riemannian manifold

Suppose that M is a complete noncompact Riemannian manifold. What is the necessary and sufficient condition that an operator on $L^{2}(M)$ comes from a smooth kernel that itself and all of the its ...

**2**

votes

**1**answer

226 views

### Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is
$$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$
where ...

**2**

votes

**2**answers

352 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 ...

**4**

votes

**2**answers

379 views

### $C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.
I ...

**2**

votes

**1**answer

91 views

### Infinite Determinant between different Hilbert Spaces

It is well-known, that if $A = \mathrm{id} + S$ is a bounded operator on a separable Hilbert Space $H$ with $S$ trace-class, then there is a well-defined notion of determinant, e.g. in terms of the ...

**4**

votes

**1**answer

174 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

77 views

### examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set ...

**5**

votes

**0**answers

73 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...

**4**

votes

**0**answers

108 views

### Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...

**6**

votes

**2**answers

270 views

### Non strictly-singular operators and complemented subspaces

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ ...

**5**

votes

**1**answer

62 views

### Continuous section inside a family of rank-varying operators

Good morning everybody,
my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear ...

**5**

votes

**1**answer

423 views

### A problem in functional calculus

This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...

**12**

votes

**1**answer

436 views

### Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...

**0**

votes

**1**answer

306 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 ...

**11**

votes

**2**answers

584 views

### Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...

**1**

vote

**1**answer

94 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 ...

**0**

votes

**2**answers

60 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
$$
...

**-3**

votes

**1**answer

174 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

194 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

80 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

31 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

123 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

198 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 ...

**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

161 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

121 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

30 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

77 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

75 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

282 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

163 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

95 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

480 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

101 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

125 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

64 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

89 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

39 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

102 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

620 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

102 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 ...