All Questions
Tagged with fa.functional-analysis hilbert-spaces
467 questions
3
votes
1
answer
1k
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Closure of polynomials of a function in $L^2$
Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict ...
- 29.7k
4
votes
2
answers
353
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Why $\lim_{n\to+\infty}\bigg(\bigg\|\sum_{f\in F(n,d)} A_{f}^* A_{f}\bigg\|^{\frac{1}{2n}} \bigg)\;\text{exists}?$
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
For $A= (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$ (not necessary to be commuting). Why
$$...
- 60.5k
2
votes
2
answers
150
views
Approximately complemented subspaces
Definition:
Suppose $E$ is a subspace of normed space $X$. Then $E$ is approximately complemented in $X$ if for any compact subset $K$ of $E$ and any $\epsilon>0$ there is a continuous linear ...
- 25.8k
5
votes
1
answer
607
views
Does eigenvalue exist in a Hilbert space? [closed]
In a lecture on Quantum mechanics, the professor concluded that if $a$ is a linear operator with $[a, a^\dagger] = 1$, where $a^\dagger$ is the adjoint of $a$ and $[a, a^\dagger] = aa^\dagger - a^\...
- 23k
4
votes
1
answer
128
views
Closure of polynomials in $L^2_w$ with log-normal weight function
Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution
$$ w(x)=\frac{1}{\...
CommunityBot
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1
vote
0
answers
127
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A point in Ion Suciu's paper on semigroups of isometric operators
My question is concerned a point in this 1968 paper by Ion Suciu which is given in Theorem 2. In the last paragraph of page 104, it is claimed that $N$ (given in the formula 2.5) is a wandering ...
- 63.9k
4
votes
2
answers
433
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A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 63.9k
1
vote
1
answer
200
views
The intersection of closure of span of infinite, linearly independent, closed, bounded, separated subsets of $\ell^2$
Let $X$ and $Y$ be two subsets of $\ell^2$ space over $\mathbb{C}$ such that: $X \cup Y$ is linearly independent, $X \cap Y = \emptyset$ and $\inf_{x \in X, y \in Y} \| x-y \|>0$ and such that each ...
- 63.9k
3
votes
1
answer
170
views
inequality involving tuple of operators on Hilbert spaces
Let $E$ be a complex Hilbert space.
Let $(A_1,...,A_n) \in \mathcal{L}(E)^n$, could you please help me to show that
$$\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{i=1}^n\| A_i^*x\|^2\bigg)\...
- 1,154
5
votes
1
answer
1k
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Space of compact operators defined on separable Hilbert space
Let $X$ be a separable Banach space and consider $\mathcal{K}(X)$ the space of compact operators $K\colon X \rightarrow X$. Is it true that the space $\mathcal{K}(X)$ is separable? If yes, why? If no, ...
- 11.3k
3
votes
1
answer
97
views
Hypercontractions and automorphisms of the unit disc
Recall that an bounded operator $T$ on a Hilbert space $\mathcal H$ is said to be $n$-hypercontraction for $n\in\mathbb N$ if
$$ I- {n \choose 1} T^*T + {n \choose 2} {T^*}^2T^2-\cdots+ (-1)^{n}{n \...
- 3,984
1
vote
0
answers
233
views
Bochner integrals with values in a Hilbert $A$-module
I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are ...
- 1,903
2
votes
0
answers
210
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A Riemannian metric on the plane such that the intersection of every two discs is a disc, again
Is there a Riemannian metric on $\mathbb{R}^2$ (or a $2$ dimensional manifold) such that the intersection of every two open discs is an open disc, again?
As linear version of this question we ask:
...
- 356
3
votes
0
answers
1k
views
Inner Product on tensor product of Hilbert spaces is unique?
Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by
$\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
- 165
2
votes
1
answer
411
views
Problem of convergence of the following sequence
Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$. Let $T\in \mathcal{L}(E)$
be bounded linear operators from $E$ to $E$ and $M\in \...
- 3,984
2
votes
0
answers
350
views
Orthonormal Basis for Convex Functions
Are there any orthonormal bases for strictly convex functions $f: \mathbb{R}^n\ni x \mapsto \mathbb{R},\ x\ne y\implies f\left(\alpha x+\left(1-\alpha\right)y\right) \lt \alpha f(x)+(1-\alpha)f(y) \...
- 13.2k
2
votes
1
answer
134
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Atomicity of blocks in a Hilbert lattice
Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?
CommunityBot
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6
votes
1
answer
1k
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Schmidt decomposition on infinite-dimensional Hilbert spaces
The Schmidt decomposition theorem says:
If $H_1,H_2$ are Hilbert-spaces (for simplicity: of same dimension) and
$x\in H_1\otimes H_2$, then there exist orthonormal bases
$\alpha_i,\beta_i$ of $...
- 146
4
votes
0
answers
164
views
A modern reference for the "Intermediate Derivatives Theorem"
In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...
- 1,247
2
votes
1
answer
257
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Cardinality of the set of Boolean subalgebras of the lattice of projections on a Hilbert space
I have a simple question I've managed to get myself quite confused about.
Given a Hilbert space H, what do we know about the cardinality of
(a) the set $P(H)$ of projection operators onto $H$ (...
- 26.3k
0
votes
1
answer
112
views
A classical fact about linear operators in Hilbert spaces
Studying the formulations which arise in hybridized mixed methods (say, mixed finite element method + hybridization), I got stuck with a rigorous proof of the following simple fact:
Let $\varphi$ be ...
- 42.8k
3
votes
2
answers
471
views
inner product on matrix spaces of multivariate polynomials?
Let $H_{n,d}=\mathbb{R}_d[x_1,..,x_n]$ be the space of $n$-variate homogeneous degree $d$ polynomials, $D=D^\top\in \mathbb{N}^{m\times m}$ a symmetric $m\times m$ matrix. Consider the space $P_D$ of ...
- 8,597
2
votes
2
answers
260
views
Bounded operators leaving dense subspace invariant
Let $A$ be a C$^*$-algebra. A pre-Hilbert $A$-module $H$ is a right $A$ module with a $A$-valued inner product (which is linear in the second variable and conjugate linear in the first variable) such ...
- 481
6
votes
0
answers
240
views
Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
- 493
2
votes
1
answer
113
views
Norm of vector-valued holomorphic functions
Let $G$ be a connected simply connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space.
Q1. Which functions $F:G\to(0,+\infty)$ are such that there is a holomorphic $f:G\to H\backslash ...
- 91.8k
3
votes
1
answer
229
views
Symmetric diagonalizable operators and self-adjointness
Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?
- 24.2k
1
vote
0
answers
113
views
Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?
In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
- 11
6
votes
1
answer
321
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Derivatives of norm of vector-valued holomorphic functions
Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
- 91.8k
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 $B(0,1)$ ...
- 25.5k
2
votes
0
answers
73
views
A question on groupoids and measurable fields of Hilbert spaces
Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...
- 1,396
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 462
2
votes
3
answers
866
views
The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant
In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on:
If $ \mathbb{H} $ is an RKHS and we denote the ...
- 550
0
votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 63.9k
3
votes
0
answers
225
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Defining a trace-class operator with a Bochner integral
I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
CommunityBot
- 1
2
votes
0
answers
341
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Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
5
votes
2
answers
2k
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On the domains and extensions of unbounded operators
I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some ...
CommunityBot
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4
votes
1
answer
165
views
Scattering of relativistic particle by long-range potential
Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of ...
- 1,191
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 ...
- 42.8k
1
vote
0
answers
220
views
About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
3
votes
2
answers
815
views
How to determine if a given set of polynomials has dense linear span in $L^2([0,1])$?
Consider the following set of polynomials: $S := \{x^{2m}: m \geq 0\}$. Is there a non-zero element $f \in L^2([0,1])$ such that $\int_0^1 fx^{2m} = 0$ for each $m \geq 0$? Note that the answer is ...
- 25.8k
2
votes
0
answers
97
views
essential self-adjointess for operators that can be factorized as $TT^*$
Let $X,Y$ be Hilbert spaces, $D$ be a dense subspace of $X$, $T:D\to Y$ be a linear operator, $\tilde{D}:=T(D)$. Assume $T^*T:D\to X$ to be essentially self-adjoint and the generated semigroup $(e^{-...
- 3,388
10
votes
3
answers
1k
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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
$$
\...
- 3,388
0
votes
0
answers
73
views
Continuously varying operators defined by a strange formula
Take $2n$-tuples of bounded positive operators $x_1,\dots x_n$ and $a_1,\dots a_n$ on a Hilbert space $H$ which have zero kernel and dense image and which satisfy the condition that (1)
$$
x_1^* x_1+\...
- 1,143
2
votes
1
answer
238
views
Hilbert-irreducible Banach space
A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition:
If a subspace $Y\subset X$ satisfies the parallelogram equality, then $Y$ is necessarilly a one ...
- 54.2k
6
votes
2
answers
672
views
Holomorphy of a function with values in a Hilbert space
Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
CommunityBot
- 1
3
votes
2
answers
570
views
Matrices Representing Bounded Operators and Absolute Values
Let $A=(a_{ij})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. For every positive integer $n$, we shall denote with $A_n$ the $n \times n$ matrix $A_n=(a_{i,j})_{i,j=1}^{n}$, and if $x \...
- 640
1
vote
0
answers
94
views
Space spanned by pointwise squares of basis functions
Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$.
Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
- 8,086
1
vote
1
answer
164
views
Hilbert-Space Values SDE in terms of Basis
Suppose:
$$
dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
- 6,045
0
votes
2
answers
608
views
Strictly convex norm on an infinite-dimensional Hilbert space
Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear
operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ ...
CommunityBot
- 1
2
votes
1
answer
651
views
Some integrals with respect to a Gaussian measure on a Hilbert space
Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
