All Questions
Tagged with fa.functional-analysis hilbert-spaces
467 questions
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
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 ...
- 5,898
22
votes
3
answers
7k
views
Subspace of $L^2$ that lies in $L^\infty$
Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?
PS. This is actually a question from the real analysis qualifier. I came ...
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
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 ...
- 661
19
votes
2
answers
2k
views
Can we take a supremum over all Hilbert spaces?
In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f_n(c)=\sup\{\|P_n\dotsm ...
- 935
18
votes
2
answers
1k
views
compact-open topology on $B(H)$
In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
- 43.2k
18
votes
2
answers
1k
views
Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies $\...
- 43.2k
16
votes
2
answers
731
views
A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space
I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book &...
- 41.8k
15
votes
4
answers
2k
views
Naive questions about "matrices" representing endomorphisms of Hilbert spaces.
This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in ...
- 41.4k
15
votes
2
answers
660
views
Multiple of identity plus compact
Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
- 247
15
votes
2
answers
2k
views
What is a projective space?
Is there a "recognition principle" for projective spaces?
What categories are there with projective spaces for objects?
Background: Although the title is a nod to What is a metric space?, ...
- 26.8k
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
13
votes
3
answers
1k
views
Is the set of separable quantum states closed?
Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable).
A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
- 896
11
votes
2
answers
478
views
$x f'$ bounded by $x^2f $ and $f''$?
Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$
I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
- 177
11
votes
2
answers
2k
views
Schur's Lemma for Hilbert spaces
Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...
- 156k
11
votes
0
answers
388
views
Von Neumann Inequality in Banach spaces
It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
- 5,529
11
votes
0
answers
529
views
Contraction semigroup on Hilbert space
I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup.
(Such operators are known as maximally dissipative operators.)
...
- 43.2k
10
votes
2
answers
606
views
A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?
Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??)...
- 41.8k
10
votes
1
answer
869
views
Complement of a subspace which is a cartesian product
Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ .
Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H =
U \oplus (V\times W)$ ?
See also Perturbations of an ...
- 2,753
10
votes
1
answer
593
views
Density of smooth function in Hilbert spaces
I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...
- 28k
10
votes
3
answers
1k
views
ordered exponential of unbounded operators
Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
\...
- 43.2k
10
votes
1
answer
899
views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
10
votes
0
answers
225
views
Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
- 233
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
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(...
- 93
9
votes
2
answers
483
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 $\alpha(t)=e^{itA}$....
- 43.2k
9
votes
2
answers
2k
views
Nice Classes of Non-Closable Operators
The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd ...
- 1,411
9
votes
2
answers
516
views
Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 1,027
9
votes
1
answer
669
views
Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
- 191
8
votes
3
answers
1k
views
When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 41.4k
8
votes
1
answer
305
views
Subspaces isomorphic with quotients
Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
- 1,361
8
votes
1
answer
747
views
Strongly continuous semigroups that cannot be contractions
Let $X$ be a Banach space, and $(P_t)_{t \ge 0}$ a strongly continuous semigroup of bounded operators on $X$. Using the uniform boundedness principle, it's simple to prove that there are constants $M,...
- 29.7k
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\...
- 81
8
votes
1
answer
522
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
8
votes
2
answers
640
views
Does a random sequence of vectors span a Hilbert space?
Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
- 1,429
8
votes
1
answer
576
views
On the definition of Hilbert spaces and real structures on Hilbert spaces
Let us consider the space $L^2:=L^2(\mathbb{R}^n,\mathbb{C})$ and the associated scalar product $S(f,g):=\int f \overline g$. In distribution theory, we have a situation where we have to deal with two ...
- 506
8
votes
1
answer
393
views
A question about comparison of positive self-adjoint operators
I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...
- 1,906
8
votes
1
answer
446
views
Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
- 1,361
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 \...
- 63.1k
8
votes
1
answer
2k
views
Definitions of Hilbert Bundles
I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
8
votes
1
answer
392
views
Proving that a space is Hilbert
Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
\|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\...
- 171
8
votes
0
answers
192
views
Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
- 60.5k
8
votes
0
answers
251
views
Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'
I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
- 81
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
7
votes
3
answers
881
views
Criterion for compactness
Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero.
Must $T$ be compact?
user130903
7
votes
3
answers
1k
views
Nice orthonormal basis for L^2(Cantor set)
Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on ...
- 81
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.5k
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
- 8,512
7
votes
1
answer
938
views
What is the idea behind interpolation spaces?
I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:
Definition. Let $H$ be an $\mathbb{R}$-...
- 131