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

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3
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0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
5answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
2answers
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 ...