All Questions
467 questions
2
votes
0
answers
786
views
Hilbert triples
I apologize if this is a trivial matter for the analysts out there, but I looked this up in a number of books (e.g., H. Brezis, J. Wloka) and could not find a satisfying discussion, one that would ...
- 29
1
vote
0
answers
136
views
A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11
3
votes
1
answer
397
views
Fractional Brownian motion via Hilbert space
The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \...
- 2,703
2
votes
1
answer
969
views
Positive definite quadratic forms on Banach spaces
This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if $E(x,x)\...
- 1,731
1
vote
0
answers
198
views
Measurability of a map that takes a functional to its composition with a linear operator
Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with the ...
CommunityBot
- 1
1
vote
0
answers
94
views
Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue square-...
- 108
9
votes
2
answers
321
views
For which $f \in L^2([0,1])$ is $f^\perp \cap C^\infty$ dense in $f^\perp$?
Given $f \in L^2([0,1])$, $f \neq 0$, we can consider the orthogonal complement $f^\perp$ . The smooth functions $C^\infty([0,1])$ are dense in $L^2([0,1])$. Is the intersection $f^\perp \cap C^\infty(...
CommunityBot
- 1
4
votes
0
answers
404
views
Hilbert Schmidt Operators and the Conditional Expectation Operator
Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
- 289
0
votes
2
answers
2k
views
Spectral decomposition of compact operators
Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
CommunityBot
- 1
3
votes
1
answer
870
views
Karhunen-Loeve expansion for discrete-time process
Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...
- 34.7k
5
votes
1
answer
873
views
Besicovitch Almost Periodic Functions a subspace of what?
The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements ...
- 41.1k
1
vote
2
answers
330
views
Characterisation of compact operators
It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.
My question is ...
- 424
7
votes
1
answer
242
views
Is there a nice "minimum" of two symmetric operators?
Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt.
...
- 25.5k
4
votes
0
answers
277
views
Exterior powers and singular values on Hilbert spaces
I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...
- 6,206
2
votes
1
answer
206
views
Special form of unbounded operators on $L_2(\mathbb{R}_+, \mathcal{H})$
I have the following problem;
Fix a Hilbert space $\mathcal{H}$. Let $S \colon \mathrm{Dom}S \subset L_2(\mathbb{R}_+, \mathcal{H}) \rightarrow L_2(\mathbb{R}_+, \mathcal{H}) $ be a closed densely ...
- 42.8k
8
votes
1
answer
844
views
A doubt about the parts of the spectrum of tensor products
Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, $T:\mathcal{H}\rightarrow\...
- 4,729
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
18
votes
2
answers
1k
views
compact-open topology on $B(H)$
In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
- 181
4
votes
0
answers
112
views
status of Invariant subspace problem on Krein Space
What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
- 2,106
2
votes
0
answers
86
views
Terminology and reference question
I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is not natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. ...
- 19.6k
0
votes
1
answer
265
views
find a weak solution in an intersection of Sobolev spaces
In
using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces
the weak solution for
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was discussed,...
CommunityBot
- 1
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
9
votes
2
answers
485
views
why is this a sufficient condition for a domain to be a core of an unbounded operator?
Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|_{t=0}$ be its infinitesimal generator, so that $\alpha(t)=e^{itA}$....
- 523
5
votes
1
answer
578
views
Infimum over all vector-valued L^2 spaces
Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to ...
- 60.5k
2
votes
1
answer
178
views
Self-adjointness of a perturbed quantum mechanical Hamiltonian specified in an infinite matrix form
Consider an operator $H$ on the Hilbert space $\ell_2$ given as an infinite matrix with two pieces, one diagonal and one arbitrary:
$H_{ij}=E_i\delta_{ij}+V_{ij}$. This has a physical meaning in ...
- 188k
20
votes
12
answers
9k
views
The role of completeness in Hilbert Spaces
Why do Hilbert spaces have to be complete?
I've been studying (teaching myself about) Hilbert spaces for a while now as they have a habit of popping up in many of the papers I'm come across (I'm a ...
- 614
4
votes
2
answers
730
views
Finite dimensional approximations of operators on Hilbert spaces
Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n \...
- 23k
8
votes
1
answer
548
views
Product of commuting nonnegative operators
Let $V$ be a real vector space with an inner product and $A,B : V \to V$ linear maps which are self-adjoint nonnegative-definite, i.e. $\langle Ax,y \rangle = \langle x,Ay \rangle$ and $\langle Ax,x \...
CommunityBot
- 1
3
votes
0
answers
168
views
Deleting "weak homeomorphism" in a Hilbert space
It is well-known that there exists a homeomorphism $h$
from an infinite-dimensional Hilbert space $H$ to $H\setminus\{0\}$.
Does there exist a "weak homeomorphism" $g:H \to H\setminus\{0\}$,
that is, $...
- 14k
4
votes
1
answer
1k
views
RKHSs containing constant functions
Suppose $H$ is the reproducing kernel Hilbert space on a space $X$ with reproducing kernel $K$. If, say, $K - c$ is a positive definite kernel for some $c>0$ then $H$ contains the constant ...
- 11.4k
1
vote
1
answer
184
views
Special kind of operators
Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where $(...
- 25.3k
3
votes
1
answer
174
views
A space parameterizing the choices of orthonormal bases for a Hilbert space
Let $\mathcal{H}$ be an infinite dimensional separable (complex) Hilbert space. What is a natural space which parameterizes the choices of orthonormal bases for $\mathcal{H}$?
It seems like one ...
- 42.8k
4
votes
1
answer
461
views
On the self-adjoint part of a quasinilpotent operator
Disclaimer: this is not research-level, but I've read some non research-level questions/answers on quasinilpotent operators here, some of them involving renowned users. So I thought I'd give it a try. ...
- 2,361
2
votes
1
answer
1k
views
Coercive Symmetric Bilinear form on a Hilbert space
I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
1) Consider a continuous symmetric ...
- 882
3
votes
2
answers
6k
views
Need help understanding Riesz representation theorem for Reproducing Kernel Hilbert Spaces
I'd like some help understanding any of the following proofs of Riesz representation theorem -- whichever is simpler -- or in fact any proof of the theorem.
Proof 1: http://nfist.pt/~edgarc/wiki/...
- 60.5k
2
votes
1
answer
386
views
Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 42.8k
4
votes
3
answers
728
views
Inequality of von Neumann for more than two contractions
Good morning,
I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
- 8,842
0
votes
1
answer
795
views
Can we construct a Hilbert space where the operator following differencial operator is symmetric?
I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...
CommunityBot
- 1
0
votes
2
answers
1k
views
Weak versus strong convergence
This is my first time posting.
I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of ...
- 44.4k
2
votes
1
answer
452
views
What do we get from an euclidian affine structure ?
Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar ...
CommunityBot
- 1
3
votes
1
answer
598
views
is a non-invertible operator a boundary point of the group of invertible operators?
Good evening,
I have a question concerning non-invertible operators.
Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence ...
- 25.8k
5
votes
1
answer
3k
views
Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
- 2,378
0
votes
1
answer
261
views
Norm functionals of $B(H)$ restricted to sub ven-Neumann algebras [closed]
Let $H$ be a Hilbert space, we know that weak topology over $B(H)$, operator algebra of bounded linear operators from $H$ into $H$, is the topology generated by
$\{\langle \cdot \xi,\eta\rangle:\; \...
6
votes
3
answers
1k
views
Topological vector spaces that are isomorphic to their duals
After reviewing the (locally convex)
topological vector spaces that I know,
the only examples I could find where there is an isomorphism from the
space to its (anti)dual, are Hilbert spaces.
So my ...
- 45.7k
1
vote
0
answers
149
views
Banach spaces with simple best approximate solutions
Let $\langle V,||.||\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that ...
CommunityBot
- 1
2
votes
1
answer
901
views
Geometry of the Hilbert sphere
Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ball-intersection if ...
- 45k
28
votes
2
answers
1k
views
Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 17.1k
-1
votes
1
answer
934
views
Domain and exponential of self- adjoint operator
Let $A$ be a self - adjoint operator on a Hilbert space $\mathcal{H}$ and let $D(A)$ be its domain. If $\psi \in D(A)$ then $exp(-itA) \psi \in D(A)$ iff $A$ is bounded ?
Thank ...
- 13k
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
3
votes
1
answer
1k
views
Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Let $A$ be a self adjoint bounded linear operator with a continuous spectrum
$\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let
$E_\lambda$ be its resolution of the identity.
For ...
