All Questions
Tagged with fa.functional-analysis hilbert-spaces
467 questions
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
134
views
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?
- 21
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 ...
- 256
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
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 ...
- 14k
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
1
answer
1k
views
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 $...
- 896
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 ...
- 5,529
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?
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
views
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 ...
- 5,529
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,
- 4,135
4
votes
1
answer
411
views
Abstract Definition of a Reproducing Kernel Hilbert Space
This is a very basic question about the definition of a reproducing kernel Hilbert space (RKHS).
It seems the standard definition of a RKHS is as a Hilbert space $H$ of functions on some set $X$ ...
- 1,307
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 ...
- 305
3
votes
0
answers
225
views
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 ...
- 131
2
votes
0
answers
341
views
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
views
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 ...
- 7,746
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 ...
- 552
6
votes
1
answer
386
views
Reference Request: Vector-Valued Ito Formula
I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...
- 5,405
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
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
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
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
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 ...
- 5,339
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 ...
- 356
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 $...
- 5,405
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 \...
- 207
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 ...
- 21
0
votes
1
answer
307
views
Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators
Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...
- 195
0
votes
1
answer
217
views
Reproducing Kernel Hilbert Spaces with positive kernels
In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
- 11
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$ ...
- 21
0
votes
0
answers
252
views
Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
2
votes
1
answer
221
views
Selfadjointness of hamiltonian with 1/x potential
Let us consider the Hilbert space $L^2([0,\infty))$ and operator $H=-\frac{d^2}{dx^2} + \frac{1}{x}$ on the domain of $C^{\infty}_0((0,\infty))$ (smooth functions with compact support away from $0$).
...
- 552
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 ...
- 221
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 ...
5
votes
2
answers
310
views
Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
- 2,246
1
vote
3
answers
684
views
Norm of an operator formed using a unitary operator
Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
- 817
4
votes
0
answers
185
views
A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058
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 ...
- 395
6
votes
1
answer
1k
views
Every self-adjoint trace class operator on $L^2$ has integral kernel
I have asked this question on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...
- 437
2
votes
1
answer
183
views
Visualizing ANOVA Decomposition [closed]
Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...
- 121
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
7
votes
1
answer
2k
views
Orthonormal bases on Reproducing Kernel Hilbert Spaces
Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $...
- 577
7
votes
1
answer
343
views
Is every $C_0$ semigroup on a Hilbert space automatically a $C_0$ group on a larger space?
Let $\{T(t),t\ge 0\}$ be a $C_0$ semigroup on a Hilbert space $X$, does that exist a larger Hilbert space $Y$ such that $X\subset Y$, and $T(t)$ extend to a $C_0$ group $T'(t)$ (so $t<0$ make ...
- 879
6
votes
1
answer
1k
views
Is the sum of spectral projections a projection?
Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...
- 241
0
votes
1
answer
163
views
Norm of derivative of rank one projector
I asked this question on math.stack but I got no answer, so I try here.
Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation}
i\partial_t\phi(t)=-\Delta\phi(t)+(V*|\phi|^...
- 113