Questions tagged [nuclear-spaces]
The nuclear-spaces tag has no usage guidance.
26
questions
3
votes
0
answers
34
views
Smooth and decaying matrix coefficients
I want to talk about the matrix coefficients
$$ f(g) = \langle \pi(g) v, \alpha^* \rangle.$$
Here $(\pi, V_\pi)$ is a representation of a Lie group $G$, with $V_\pi$ being a topological vector space ...
3
votes
0
answers
76
views
"Weakly" nuclear operators (terminology)
Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name.
Let's say that a linear map $T:V\to W$ between locally convex topological ...
- 81
4
votes
1
answer
113
views
If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?
Let $F$ a Fréchet space.
This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$.
Let's ...
- 536
4
votes
1
answer
138
views
Are nuclear spaces used in creating variant theories of distributions?
Laurent Schwartz proved his Kernel Theorem in 1952 to justify extending his theory of distributions to several variables. Then he and Jean Dieudonne gave Alexander Grothendieck the assignment to ...
- 10.5k
18
votes
2
answers
684
views
What is known about the "unitary group" of a rigged Hilbert space?
Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
- 536
6
votes
3
answers
827
views
Are nuclear operators closed under extensions?
Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X_1 @>f_1>> X_2 ...
- 1,268
9
votes
1
answer
205
views
A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm
Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.
- 279
3
votes
1
answer
327
views
multiplier algebra of a simple $C^*$ algebra
If $A=K(H)$, where $H$ is an infinite dimensional separable Hilbert space, then $A$ is simple and nuclear, and the multiplier algebra $M(A)$ of $A$ is not nuclear.
My question is: can we find a non-...
- 451
1
vote
1
answer
150
views
Nuclear operators and their eigenvalues
I know that if an operator (on a Banach space with approximation property) is nuclear of order zero, then its eigenvalues are $p$-summable for any $p>0$. (I read it from Grothendieck’s book “...
- 241
7
votes
1
answer
250
views
Completely bounded maps approximately factoring through finite matrices
Let $A$, $B$ be two $C^\ast$-algebras and $\mathcal{F}(A,B)$ be the operator ideal of all completely bounded operators $T:A \to B$ for which there are uniformly bounded nets of completely bounded maps ...
- 1,040
2
votes
0
answers
65
views
Nuclear operator between general topological modules over ultrametric Banach rings
In the celebrating paper "Completely continuous endomorphisms of p-adic Banach spaces", Serre established a Fredholm-Riesz theory for compact endomorphisms of Banach spaces over (spherically complete)...
- 131
9
votes
2
answers
424
views
Traces of operators in nuclear spaces
I am currently reading up on nuclear spaces in Jarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:
Let $F$ be a nuclear ...
- 91
4
votes
1
answer
296
views
Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$
Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...
- 1,483
8
votes
1
answer
459
views
Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?
Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial.
For $m\in \mathbb{Z}$, we also define $\...
- 2,082
13
votes
1
answer
1k
views
Error in Maurins proof for the nuclear spectral theorem?
I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or ...
- 408
3
votes
2
answers
332
views
Linear operators on distributions with different topologies
Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of $\mathscr{D}...
- 9,147
1
vote
0
answers
188
views
Reference for a General Theory of Sequences?
Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis.
In that works, sequence spaces are generally ...
- 2,082
10
votes
2
answers
1k
views
Wiener measure and Bochner Minlos
I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...
- 341
10
votes
3
answers
698
views
Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
- 8,132
6
votes
2
answers
2k
views
What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?
Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions.
We consider $\...
- 2,389
18
votes
7
answers
6k
views
Grothendieck on Topological Vector Spaces
In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he told Bernard Malgrange ...
5
votes
1
answer
740
views
Schwartz Kernel theorem for tempred functions
Let $T(R)$ denote the space of tempered functions on the line,
i.e. the smooth functions that give Schwartz function after a
multiplication by any Schwartz function, equipped with the natural
nuclear ...
- 2,389
8
votes
1
answer
813
views
Exactness of completed tensor product of nuclear spaces
Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence
of (complete) nuclear spaces, i.e. it is a short exact sequence of
(complete) nuclear spaces, all the maps are continuous, the map $...
- 2,389
2
votes
0
answers
303
views
Hom of Nuclear spaces
Let V be a Nuclear space (not necessary Frechet, but complete). Let V^* be the dual space considered with the weak topology (i.e. pointwise convergence). Is it true that $V^*$ is also nuclear?
Is it ...
- 2,389
16
votes
0
answers
998
views
Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...
- 32.5k
15
votes
2
answers
717
views
Are extensions of nuclear Fréchet spaces nuclear?
Consider the category of Fréchet spaces, the morphisms being
continuous linear maps with closed image. Suppose that we
have a short exact sequence in that category:
$0 \rightarrow V_1 \rightarrow ...
- 241