For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

learn more… | top users | synonyms

3
votes
0answers
38 views

Under what conditions is the resolvent set of a linear operator connected?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$. As usual, we define ...
0
votes
0answers
24 views

Show that $l^2(\Bbb N , F)$ with equipped norm is not complete.

For $v$ in $l^2(\Bbb N , F)$ with norm on $l^2$ defined as: $\lvert\lvert v\rvert\rvert_{W}= \sum^\infty_{k=1}\frac{\lvert v_{[k]}\rvert}{2^k}$ Show that $l^2(\Bbb N , F)$ with the norm ...
2
votes
1answer
68 views

Every partially defined isometry can be extended to a isometry

I know that the following theorem holds true: Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual ...
4
votes
1answer
58 views

Understanding the definition of the covariance operator

Let $\mathbb H$ be an arbitrary separable Hilbert space. The covariance operator $C:\mathbb H\to\mathbb H$ between two $\mathbb H$-valued zero mean random elements $X$ and $Y$ with $\operatorname ...
1
vote
0answers
26 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
0
votes
0answers
39 views

A question about Hilbert Spaces and convex sets

I am struggling with this and could really do with some help: Let $H$ be a Hilbert space over $\mathbb{R}$, $\{v_n\}$ be a sequence of vectors in $H$, and $C$ be a convex subset of $H$ containing ...
15
votes
4answers
1k views

What really is ''orthogonality''?

I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real ...
2
votes
0answers
24 views

Hilbert space and traces

Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define ...
0
votes
1answer
40 views

Homogeneous and Inhomogeneous Function Spaces

I would like a general explanation on the difference between homogenous and inhomogeneous function spaces, there doesn't seem to be a very good explanation online. I know that for Sobolev spaces for ...
1
vote
3answers
45 views

Is faithful positive sesqulinear form an inner product?

As in the title: does a positive sesqulinear form need to be conjugate-symmetric? Background: The question comes from an attempt to understand the proof of the Stinespring representation theorem. It ...
0
votes
1answer
29 views

Norm of a self adjoint operator

Let $T$ be a (bounded) self-adjoint operator on a Hilbert space. Is it true that $||T^k|| = ||T||^k$ for all positive integers $k$? It's true for $k=1,2$, and I'm wondering if this could be ...
2
votes
2answers
37 views

Why can't a Hilbert curve be used to put the real numbers into a listable format?

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ...
10
votes
1answer
182 views

When is a function of the largest eigenvalue continuous and/or differentiable?

I want to understand why the following function, the largest eigenvalue of a symmetric linear operator, is continuous and Gâteaux differentiable. \begin{equation*} \lambda(V)=\sup_{f \in \ell^2(I):\ ...
2
votes
1answer
30 views

Let $A$ be a non-separable $C^*$-algebra. Is it possible that there is a faithful representation $\pi:A\to L(H)$ on a separable hilbert space $H$?

Let $A$ be a non-separable $C^*$-algebra. Is it possible that there is a faithful representation $\pi:A\to L(H)$ on a separable hilbert space $H$? I know that if $A$ is separable, one can choose $H$ ...
1
vote
1answer
35 views

Does an essentially self-adjoint operator have the same kernel as its closure?

Let $H$ be a Hilbert space and let $A : D(A) \subset H \to H$ be an essentially self-adjoint operator. Let $\overline A$ be the unique self-adjoint extension of $A$. Question: Is it true that ...
0
votes
1answer
32 views

definition of block diagonal operator on a hilbert space

I 'm stuck with the definition of block diagonal operators on hilbert spaces. Def.: A bounded linear operator $T$ on a hilbert space $H$ is called block diagonal if there exists an increasing ...
2
votes
0answers
27 views

Idempotent and positive definite operator implies self adjoint

Let $H$ be a Hilbert Space (over $\mathbb{R}$ or $\mathbb{C}$ but maybe is valid for any field) and $E$ a continuous operator. Suppose $E$ is idempotent, i.e.,$E^2=E$ and positive definite, i.e. ...
2
votes
2answers
53 views

What is the $C^*$-algebra generated by a normal operator?

The following is the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I don't find the definition for the $C^*$-algebra generated by a normal operator in the book. ...
0
votes
1answer
47 views

Weak Solutions to PDES

I am working through some practice problems for my PDE class and I came across the following: Let $U\in \mathbb{R}^n$ be a smooth, bounded, connected open set. Let $\Gamma_1$, $\Gamma_2$, be two ...
4
votes
3answers
44 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
1
vote
1answer
41 views

Projection Theorem

I've been trying to apply the projection theorem to the following problem with no success. I've spent a few hours on this today, any help would be appreciated. Let H be a finite dimensional Hibert ...
2
votes
3answers
102 views

Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up

I have two questions regarding how two concepts that involve complex valued functions may come up in a natural way. (Non-natural ways are: These concept come up in order to present a unified theory, ...
2
votes
0answers
44 views

How can we use theory from $L^2(\mathbb{R})$ on a sequence of numbers (discrete signal)

In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal. look at this paper ...
1
vote
1answer
53 views

Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
3
votes
1answer
35 views

Flat Extrinsic Vs. Intrinsic Distance

Context: Let $\Psi: \mathbb{R}^d \rightarrow \mathscr{H}$ be a $C^k$-embedding of $\mathbb{R}^d$ into a Hilbert space $\mathscr{H}$. We may view $\mathscr{M}:=Im(\Psi)$ as a submanifold of ...
4
votes
1answer
44 views

Direct sum decomposition of $L^2(\mathbb{R})$ using Fourier Transform

Let $L_+^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^+}\}$ and $L_-^2(\mathbb{R})=\{f\in L^2(\mathbb{R}):supp \hat{f}\subset\mathbb{R^-}\}$, where $\hat{f}$ denotes the Fourier ...
0
votes
1answer
23 views

If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
0
votes
0answers
27 views

If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
1
vote
0answers
25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for ...
3
votes
1answer
22 views

Is a bounded, linear, nonnegative and symmetric operator with finite trace on a Hilbert space Hilbert-Schmidt?

Let $U=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
0
votes
0answers
10 views

Property related to weak convergence

Let $H$ be a real Hilbert space and $F:H\rightarrow H$ satisfying $$ \lim\langle F(u_k),v_k-u_k\rangle=\lim\langle F(u),v-u\rangle, $$ for all sequences $\{u_k\},\{v_k\}$ such that $u_k$ converges ...
3
votes
1answer
34 views

norm of orthogonal projection of some vector in Hilbert space

Let $H$ be Hilbert space and $u_1,u_2,...u_n \in H$ (vectors dont have to be orthogonal) $V=span\{u_1,u_2,...u_n\}\subset H$ and $S$ is unit sphere in $V$. $P_V$ is orthogonal projection on V. Now ...
0
votes
0answers
21 views

Orthonormal Basis of Hermitian matrices for a Hilbert Space of Operators

Consider a set of operators $O$ on a Hilbert space $V$ of dimension $d$. I could prove that $O$ is also a Hilbert space with dimension $d^2$ (inner product being $(A,B) = tr(A^\dagger B))$. Now I am ...
1
vote
1answer
43 views

How does this argument show continuity?

I want to show that the unitary group $U(\mathcal H)$ of a Hilbertspace $\mathcal H$ is a topological group wrt the strong operator topology. For the standard proof it is most convenient to use that ...
2
votes
1answer
66 views

The strong topology on $U(\mathcal H)$ is metrisable

The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the ...
0
votes
1answer
32 views

Relation between complete spaces and usage of calculus

I have been studying about metric spaces and completion of metric spaces. While reading into Hilbert spaces, I discovered this phrase on their Wikipedia webpage:" Hilbert spaces are complete: there ...
0
votes
1answer
51 views

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis

Using direct sums, construct an inseparable Hilbert space with an uncountable orthonormal basis. This is Problem 13 in Chapter II in Reed & Simon, and I'm really stuck on this one. Would ...
0
votes
0answers
24 views

Meaning of completeness for a sequence

I am studying some course notes and came across the following proposition: "An orthonormal sequence in a Hilbert space H is complete iff the only vector in H which is orthogonal to each of the ...
1
vote
0answers
25 views

An application of the Closed Graph Theorem [duplicate]

Let $T:L^2([0,1]) \to L^2([0,1])$ be a bounded linear map of Hilbert spaces such that if $f\in L^2([0,1])$ is continuous then so is $Tf$. Show that there is a positive constant C such that ...
2
votes
1answer
20 views

Extension of Lipschitz Continuous Operators on arbitrary sets in Hilbert Spaces

Let $T: D(T) \subset X \to Y $ be a Lipschitz continuous operator on an arbitrary set $D(T)$ in the Hilbert Space $X$. Show that $T$ can be extended to an operator $\tilde{T}:X \to Y$ which is ...
3
votes
2answers
30 views

Is the range of a self-adjoint operator stable by its exponential?

Let $H$ be an Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $A$. We assume that $R(A)$, the range of $A$, is not closed. Is it true or not that $R(A)$ is stable by ...
2
votes
1answer
50 views

If $Q$ is an operator on a Hilbert space with $Qe_n=λ_ne_n$ for all $n$, then $Q^{-\frac 12}e_n=\frac 1{\sqrt{λ_n}}e_n$ for all $n$ with $λ_n>0$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\mathfrak L(U)$ be the set of bounded and linear operators on $U$ $Q\in\mathfrak L(U)$ be nonnegative and symmetric ...
0
votes
0answers
17 views

Are spherical harmonics a basis for $H^1$?

We know that spherical harmonics are a complete orthonormal system for $L^2(\mathbb{S}^2)$. Is it true that they are also a complete orthonormal system for $H^1(\mathbb{S}^2)$? Furthermore, is it ...
0
votes
1answer
39 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
1
vote
0answers
37 views

What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where ...
2
votes
1answer
47 views

Proof of equivalence of $\lambda$ norms in Sobolev space $H_0^1(\Omega)$

Consider the following metrics in $H_0^1(\Omega)$ with $\Omega$ a bounded domain: $$\| u\|_\lambda=\left( \int_\Omega |\nabla u|^2+\lambda\int_\Omega u^2\right)^{\frac{1}{2}}$$ and $$\| u\|_0=\left( ...
0
votes
0answers
17 views

Explanation of the proof of Theorem 2.13 in Young, “An introduction to Hilbert Space”

Let $\lVert\cdot\lVert$ be any norm on the vector space $E$ and let $\rho\left(\sum^n_{j=1}\lambda_je_j\right)=\left(\sum^n_{j=1}|\lambda_j|^2\right)^{1/2}$ where $(e_j)$ is a basis for $E$. Now ...
1
vote
0answers
28 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
1
vote
0answers
33 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
2
votes
0answers
32 views

Given a linear Hilbert-Schmidt embedding $ι$ between Hilbert spaces, prove that $ιι^*$ is a bounded, linear operator with finite trace

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $U_0:=Q^{\frac 12}(U)$, $$\langle u,v\rangle_0:=\langle ...