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

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1answer
66 views

For a Hilbert space $\mathcal{H}$, is every bounded linear operator on $\mathcal{H}$ a linear combination of unitary operators?

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space, and let $B \in \mathcal{B}(H)$ be a bounded linear operator on $H$. If $\mathcal{H}$ is a complex Hilbert space, then $B$ can be written as a ...
2
votes
1answer
23 views

Proving a particular subset of a Hilbert space is a subspace

I have a small question please, how to prove that this set: $F=\lbrace h\in H, \langle f''(u)h,h\rangle <0\rbrace$ is a sub space of the Hilbert space $H$, where $f''(u)$ is a self-adjoint ...
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0answers
33 views

Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
1
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1answer
42 views

Prove that a Hilbert space is convex of power type $2$

Let $X$ be a Banach space. For $\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that $X$ ...
0
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0answers
22 views

Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
1
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1answer
29 views

Prove that a linear and continuous operator admits inverse in Hilbert space

Let $(H,(\cdot,\cdot))$ an Hilbert space and $A:H\rightarrow H$ a linear and continuous operator such that there exists $\alpha >0$ such that $$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$ ...
1
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1answer
31 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
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1answer
95 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
0
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1answer
95 views

Square Summable functions

Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks.
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0answers
47 views

Prove that operator is completely continuous

Let's consider Banach space $\ell^\infty$ of bounded sequences $x = \{ \xi_n\}_{n=1}^\infty$: $$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$ Suppose matrix $||a_{i j}||_1^\infty$ specifies operator $A$ ...
0
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1answer
31 views

Weakly sequentially continuous operators in Hilbert space are norm continuous.

Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous. I feel that this statement ...
0
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1answer
35 views

Riesz (Hilbert-space) representation theorem and dirac delta on $\mathcal{C}_{0}$

I am thinking about this for a while now, but don't get near an understanding, so I must have gotten something important wrong. I look at $\mathcal{C}_{0}$, the space of countinuous (bounded) ...
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1answer
43 views

Relation between $\epsilon$-pseudospectrum of operators

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why ...
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1answer
77 views

How to find all isometries of Hilbert space?

We know all isometries of $\mathbb R^n $ are composition of transfer by orthogonal linear functions. How to find all surjective isometries of Hilbert space? Is there similarity?
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0answers
44 views

difference between uniformly convex norms and strictly subadditive norms?

What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by ...
0
votes
1answer
66 views

Formulas for Schrödinger unitary groups of operators

Let $\Omega$ an open set of $\mathbb{R}^n$. Consider the Hilbert space $X=L^{2}\left(\Omega\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\Omega)$. Is there any ...
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3answers
64 views

Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
2
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1answer
43 views

Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
2
votes
1answer
55 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
3
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2answers
114 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
5
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1answer
96 views

Show that a subspace of l2 is not complete

I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ...
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1answer
32 views

How to prove the 'uniform summability' of a Cauchy sequence?

I have an exercise given by the teacher and I'm pretty sure that this proof is not hard, but I don't have idea how to approach it. I have to prove the 'uniform summability' (this name was used by ...
3
votes
2answers
79 views

Properties of a set in $\ell^2$ space

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$ and $e_n \in \ell^2 $ be the sequence whose $n$-th element is 1 and all other elements ...
0
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2answers
64 views

Strongly continuous semigroup of operators which cannot be extended to a group

Let $X$ be a Banach space. We call a family of bounded operators $(T(t))_{t\in \mathbb{R}}$ a strongly continuous group if it satisfies the properties of the strongly continuous semigroup but for ...
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1answer
22 views

Spectrum of adjoint bounded linear operator on hilbert space

I have been struggling to analyse the spectrum of the adjoint of a bounded linear operator on a hilbert space. Throughout the internet I have found vague references that $\sigma(T^*) = \sigma(T)$ but ...
2
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0answers
47 views

Prove $|(f, g)| \leq \int |f \bar g|$ for Complex Cases

Let $f, g$ be $\mathbb C$-valued functions defined on $\mathbb R$ and $f, g \in L^2$. To prove the inequality in this title, I proceed as follows but got a weaker bound. Recall that $\mathrm{Re}\ a ...
2
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2answers
127 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
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0answers
28 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
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1answer
35 views

Is there exists linear algebra basis for $L^2[0,1]$ such that every element of it has length one and every two different element of it is orthogonal?

We know by using axiom of choice every vector space over a division ring ( consequently any field ) has a basis like $\mathbb E$ in the meaning of linear algebra ( $\mathbb E$ is linear independent ...
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0answers
41 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
4
votes
1answer
145 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
0
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1answer
21 views

Orthonormal basis $L^2(a,a+2\pi)$

Let $$\mathcal{B}=\left \{\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt{\pi}},\frac{\sin x}{\sqrt{\pi}},\frac{\cos 2x}{\sqrt{\pi}},\frac{\sin 2x}{\sqrt{\pi}},\dots\right \}$$. This is an orthonormal basis ...
1
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1answer
42 views

Weak convergence plus strong convergence

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to ...
2
votes
2answers
97 views

Can Hilbert spaces generalize non-Euclidean geometry by having the sum of the angles of a triangle not be equal to pi?

I am an amateur mathematician learning new things. Let A and B be vectors in a Hilbert space. The three vectors A, B and A-B form a triangle. The idea of the angle between two vectors can be ...
2
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1answer
67 views

Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
2
votes
1answer
36 views

Weak convergence and infinite sum

Suppose that $\psi_n$ converges weakly to $\psi$ in a Hilbert space $H$. Assume further $\{\phi_k\}$ is an orthonormal sequence in $H$. Is it plausible that ...
2
votes
1answer
63 views

Is a linear operator on $\ell^2$ defined by the inner product necessarily bounded? [duplicate]

If $a=\{a_n\}\in \ell^\infty(\mathbb{R})$ and $\langle a,x \rangle<\infty$ for all $x\in \ell^2(\mathbb{R})$, (where $\langle a, x\rangle=\displaystyle \sum_{k=1}^\infty a_kx_k$), then is $a\in ...
2
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0answers
44 views

Discontinuous linear operator on $\ell^{2}$

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear ...
0
votes
1answer
30 views

Linear operator on a hilbert space

Let $\{e_{n}\}_{n = 1}^{\infty}$ be an orthonormal basis of a Hilbert space $H$. Suppose $T: H \rightarrow H$ is a linear operator. For each $x \in H$, then $x = \sum_{n}\langle x, e_{n}\rangle e_{n}$ ...
0
votes
1answer
45 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
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1answer
64 views

Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
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0answers
20 views

Compactness of the projection operator

Let $H$ be a Hilbert space and let $F$ be a closed subspace. Then I'm to prove that the projection $p:H\rightarrow F$ is a compact operator if and only if $F$ is finite-dimensional. It's easy to prove ...
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0answers
26 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
0
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1answer
30 views

Linear operator defined by its eigenvectors/values

Let $H$ be a Hilbert space, $(e_n)$ a complete orthonormal sequence, and $\lambda_n$ a bounded sequence of complex numbers. Let $A$ be defined such that the $(e_n)$ are the eigenvectors of $A$ and the ...
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1answer
54 views

Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
0
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0answers
23 views

Symmetricity of square-integrable functions respect to inner-products.

Let $f(x),g(x) \in \mathrm{L}_2$. If we define inner-product $(f(x),g(x)) = \int_a^b f^*g dx$, then prove that $(f,pg) = (pf,g)$, given $f(1) = Cf(0), C=e^{i\psi}, p=-i d/dx, a=0, b=1$. I tried ...
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1answer
27 views

Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
0
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1answer
45 views

Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.

Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...
4
votes
1answer
73 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
3
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1answer
45 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb ...