All Questions
Tagged with fa.functional-analysis hilbert-spaces
467 questions
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,...
- 63.9k
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.)
...
- 63.9k
4
votes
0
answers
2k
views
Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 42.8k
0
votes
0
answers
191
views
Canonical embedding of Hilbert space in random $L^2$
This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...
- 1,329
2
votes
1
answer
315
views
Estimate the metric entropy of unit ball in $L^2$ space
Let me clarify the setting I'm thinking.
For any totally bounded metric space $(Y,d_Y)$ and $\varepsilon>0$, the $\textit{metric entropy}$ $N_M(\varepsilon,Y)$ is the smallest number of closed ...
- 109k
0
votes
1
answer
421
views
Canonical embedding of Hilbert space in $L^2$ space
Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-...
- 128k
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
4
votes
2
answers
313
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
- 692
1
vote
1
answer
629
views
What is the difference between $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$? [closed]
$L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$ are isometrically isomorphic because both are infinite-dimensional separable Hilbert spaces.
If a Hilbert space $H$ is $L^2(\mathbb{R})$ or $L^2(\mathbb{R}^2)...
- 42.8k
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 ...
- 1,361
2
votes
0
answers
55
views
A holomorphic map into a Hilbert space with prescribed orthogonality
This is a variation of my previous question.
Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
- 5,529
4
votes
1
answer
174
views
A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 61.9k
3
votes
0
answers
149
views
What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?
Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...
- 2,041
5
votes
1
answer
334
views
Iterated limits equal?
Consider the Banach algebra $B_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$ with Hilbert Schmidt norm. We know that $B_2(H)$ is a Hilbert space as well with $\left<A,B\right>=tr(B^*...
- 18.7k
0
votes
0
answers
113
views
Is it possible that the dimension of the intersection of a nested sequence of Hilbert space is 1?
Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$
Let $\{h_n\}_{n \in \mathbb{N}} \in H$ be a sequence of linearly independent vectors in $H$
Let
$$
V=
\bigcap_{n=1}^\...
- 433
0
votes
1
answer
675
views
Weak convergence of Hilbert Schmidt operators
So I am stuck at this situation. Let $\{A_n\}$ be a weakly convergent sequence in $B_2(H)$ converging to $0$ in the weak topology on $B_2(H)$. Which means that $\left<A_n,D\right>=\operatorname{...
1
vote
0
answers
80
views
Measurability of a generalized point spectrum
Assume that $ T:H\oplus H\rightarrow H\oplus H$ is a unitary linear operator on the double sum of a separable Hilbert space $H$ with itself.
Let us call a pair $(\lambda, \mu)\in\mathbb{C}\oplus\...
- 728
1
vote
1
answer
2k
views
interchanging limits and summation
So I am stuck at this situation . Suppose $m:B_2(H_1)\times B_2(H_2)\to \mathbb C$ be bilinear form given by $m(S,T)=\left<T,\phi(S)\right>$, where $\phi: B_2(H_1)\to B_2(H_2)$ be a bounded ...
- 42.8k
3
votes
1
answer
148
views
Is this sum of nonexpansive maps itself nonexpansive?
For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix
$...
- 12.5k
1
vote
0
answers
73
views
Domain of definition of a hamiltonian with delta(contact) potential
I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with $\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$)
$$...
- 111
4
votes
1
answer
110
views
Graded adjointable operators on a graded Hilbert space
Given a graded Hilbert space $\mathbf{H} = \bigoplus_{k \in \mathbb{N}} \mathbf{H}_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to ...
- 18.7k
-1
votes
1
answer
265
views
A sequence of Hilbert spaces and a sequence of linear funtionals [closed]
Let $H$ be an Hilbert space over $\mathbb{C}$
Let $\{h_m\}_{m \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$
Let $\forall m \in \mathbb{N}: H_m = \overline{\...
CommunityBot
- 1
1
vote
0
answers
53
views
Spectrum of a $1$-parameter family of symmetric linear operators
I am working with certain submanifolds of symmetric spaces and, using a construction in Terng-Thorbergson, we ended up in the following Hilbert space problem:
Let $H$ be a (real) Hilbert Space and $...
- 163
6
votes
1
answer
119
views
Tensoring adjointable maps on Hilbert modules
Given a right Hilbert $A$-module $E$, and a right Hilbert $B$-module $F$, together with non-degenerate $*$-homomorphism $\phi:A \to \mathcal{L}_B(F)$, we can form the tensor product
$$
E \otimes_{\phi}...
- 18.7k
2
votes
1
answer
917
views
Characterising closed range self-adjoint operators
Let $T:\mathrm{dom}(T) \subseteq H \to H$ be a densely defined, self-adjoint operator on a Hilbert spaces $𝐻$. In general the range of $T$ is not guaranteed to be closed. What tools are available to ...
- 18.7k
2
votes
1
answer
107
views
tensor stability of block-positive matrices
Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...
- 18.7k
5
votes
2
answers
642
views
Non-standard tensor products of inner product spaces
For two inner product spaces $(\mathcal{V}, (\cdot,\cdot)_V)$ and $(\mathcal{W}, (\cdot,\cdot)_W)$, we can put an inner product on their tensor product in the obvious way:
$$
(1) ~~~~ \langle v \...
- 643
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
3
votes
1
answer
285
views
Closable unbounded operators and Banach space adjoints
For an unbounded operator $T:\mathcal{H}_1 \to \mathcal{H}_2$, if its adjoint $T^*$ is densely defined, then we know that $T$ is closable. What happens if we replace $\mathcal{H}_1$ or $\mathcal{H}_2$ ...
- 18.7k
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 ...
- 17.2k
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 95
5
votes
1
answer
189
views
Intersection of spaces with Schauder basis
Let $\{v_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $V$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $V_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m}$
Let $\{...
- 60.5k
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
0
votes
1
answer
294
views
If $A$ is a dissipative self-adjoint operator with spectral decomposition $(H_λ)$, then $e^{tA}x$ tends to the projection of $x$ onto $H_0$ as $t→∞$
Let $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with dissipative self-adjoint generator $(\mathcal D(A),A)$. In particular, $T(t)$ is self-...
- 167
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.
$$
It satisfies $\...
- 43.2k
4
votes
1
answer
151
views
Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
- 10.1k
5
votes
1
answer
465
views
Quasinilpotent , non-compact operators
If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer ...
- 31.5k
0
votes
3
answers
501
views
The completion of $F/\text{Ker}(M)$ is isomorphic to the closure of the range of $M$
Let $M$ be a positive semidefinite operator on a complex Hilbert space $(F,(\cdot,\cdot))$.
On the quotient space $F/\text{Ker}(M)$ we have the following inner product
$$\langle \overline{x},\...
- 16.4k
1
vote
0
answers
277
views
Adjoint for a non-densely defined unbounded operator on a Hilbert space
Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
- 993
2
votes
1
answer
399
views
A sequence of orthogonal projection in Hilbert space
Let $H$ be an infinite dimensional Hilbert space over $\mathbb{C}$
Let $\{v_n\}_{n \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ such that $v_n \to u$
Let $\forall ...
- 12.5k
3
votes
0
answers
171
views
Decomposition of $L^2(\Gamma \backslash H)$ into irreducible representations using the spectral theorem
I'm reading the introduction of An Introduction to the Trace Formula by James Arthur and wanted to understand something in the introduction.
Let $H$ be a unimodular locally compact Hausdorff group, ...
CommunityBot
- 1
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 (??)...
CommunityBot
- 1
2
votes
1
answer
247
views
Intersection of Hilbert spaces with basis equivalent to canonical basis for $\ell^2$
Let $X$ be a closed subspace of $\ell^2$ over $\mathbb{C}$
Let $\{x_m\}_{m \in \mathbb{N}}$ a basis for $X$ equivalent to canonical basis $\{e_m\}_{m \in \mathbb{N}}$ for $\ell^2$
I Would like to ...
- 42.8k
4
votes
1
answer
127
views
Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator
$$
\begin{aligned}
&\Gamma_0\...
- 4,361
3
votes
0
answers
495
views
Simple (?) question on inner product in reproducing kernel Hilbert space
I'm following the gentle introduction to Reproducing Kernel Hilbert Spaces From Zero to Reproducing Kernel Hilbert Spaces in Twelve Pages or Less by Hal Daumé III. I believe the author fully ...
- 63.9k
0
votes
1
answer
80
views
Vectors concentrated on one coordinate
Suppose $X$ is a Banach space, $(e_i)$ a normalized basis, $(e_i^*)$ the biorthogonal functionals, and $Y$ a finite codimensional subspace of $X$. Given $N$ and $\varepsilon$, can we find $x\in Y$ ...
- 109k
2
votes
1
answer
397
views
Can a bijection between function spaces be continuous if the space's domains are different?
It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...
- 902
6
votes
0
answers
113
views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
- 395
7
votes
1
answer
311
views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
- 9,486