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

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5
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1answer
78 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
2
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2answers
38 views

Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
1
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1answer
13 views

Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
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0answers
30 views

Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.
2
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0answers
30 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
3
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1answer
32 views

Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$. For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, ...
2
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0answers
42 views

A matrix as eigenvalue?

I wonder if some work has been developed on operators in Hilbert space that have the property of having matrices instead of numbers as eigenvalues (the matrices do not necessarily act on vectors in ...
2
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1answer
26 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
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0answers
36 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
2
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1answer
45 views

Chain of closed subspaces in a Hilbert Space

Let $H$ be a separable Hilbert Space (WLOG, we may assume $H=\ell_2(\mathbb{N})$ is the space of square summable sequences). Can there exist an uncountable chain of closed subspaces? In other words, ...
2
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1answer
36 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...
4
votes
1answer
47 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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1answer
35 views

What is the difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
2
votes
1answer
22 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
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0answers
69 views

Conditions for the commutator of two operators on a Hilbert space to not be a nonzero scalar operator

I have shown a proposition: Suppose $A$ and $B$ are two linear operators on a (complex) Hilbert space, where the domains may not be the whole. Then, if either $A$ or $B$ is normal and has an ...
2
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1answer
33 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
4
votes
1answer
44 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
31 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.
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1answer
36 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 $X$ ...
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0answers
16 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 ...
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1answer
28 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 $$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$ ...
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1answer
25 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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0answers
49 views

Inseparable Hilbert space and uncountable orthonormal basis construction

I need help with exercise 13 from Methods of Modern Mathematical Physics I by Simon and Reed, chapter 2. Using direct sums, construct an inseparable Hilbert space and an uncountable orthonormal ...
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1answer
32 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 ...
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1answer
25 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
43 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
23 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
21 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
37 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
73 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?
2
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0answers
26 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
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1answer
64 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
59 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
41 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
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1answer
51 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
60 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
67 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
28 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 ...
2
votes
2answers
67 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 ...
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2answers
44 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
19 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
37 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 ...
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2answers
83 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
23 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
30 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
39 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
101 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 ...
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1answer
35 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 ...