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

learn more… | top users | synonyms

1
vote
0answers
37 views

Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
0
votes
0answers
24 views

Relation between two spectra

This seems like an easy enough computation but I'm stuck! Let $X \in B(H)$ for a Hilbert space $H$ such that $X^{2}=0$, but $X\neq 0$. With respect to the decomposition $H=\text{ker}X \oplus ...
2
votes
1answer
46 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
1
vote
1answer
45 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
2
votes
0answers
18 views

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$. (HINT: The series $$\sum_n c_n e^{i\lambda_n t}$$ with $\lambda_n=n-1/4$, diverges in ...
2
votes
1answer
31 views

Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
3
votes
1answer
27 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
2
votes
1answer
32 views

Prove that the space P is a Hilbert Space.

Prove that the space P of all entire functions of the form $$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(t) e^{-izt} dt,$$ is a Hilbert Space, where $\varphi\in L^2[-\pi,\pi]$. The inner product of ...
2
votes
0answers
42 views

Distance to a closed subspace of a Hilbert space in terms of inner product with the unit normal

Let $M$ be a closed subspace of a Hilbert space $H$, and suppose $x_0\in H$ Show that: $$\min(\|m-x_0\|, m\in M)=\max(|\langle x_0,n\rangle|, n\in M^\perp ,\|n\|=1)$$ I know that $|\langle ...
1
vote
0answers
16 views

Does a set of 'm' linearly independent continuous functions constitute a Hilbert Space

If I have a Sobolev space $\mathcal{H}^m[a,b]$ of functions $f : [a,b]\rightarrow\mathbb{R}$ where for all $f \in\mathcal{H}^m[a,b]$, $f$ and all derivatives up to order $m-1$ are absolutely ...
1
vote
1answer
41 views

Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally ...
1
vote
1answer
38 views

Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
1
vote
1answer
32 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
0
votes
2answers
49 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
3
votes
2answers
51 views

How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
0
votes
0answers
42 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
0
votes
1answer
22 views

Continuous Quadratic Form $\implies$ Continuous Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form $q:\mathcal{H}\to\mathbb{C}$. Define its inducing sesquilinear form: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{C}: ...
0
votes
1answer
21 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
0
votes
1answer
40 views

Isomorphism between Euclidean space and its conjugate

I know that, if $H$ is a Hilbert space, for any continuous linear functional $f\in H^{\ast}$ there is a unique element $x_0\in H$ such that $\forall x\in H\quad f(x)=\langle x,x_0\rangle$. Moreover, ...
1
vote
3answers
43 views

Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
2
votes
1answer
40 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
3
votes
1answer
39 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
1
vote
0answers
16 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
0
votes
3answers
60 views

Selfadjoint Operator: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
1
vote
2answers
31 views

Can we show that $E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$

Let $X,Y,Z$ be some random elements on some Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Can we show that $$E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$$ I can clearly see that $$E\|X-Y\|^2 \leq ...
5
votes
1answer
91 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
votes
2answers
43 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
vote
1answer
17 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 ...
0
votes
0answers
34 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
votes
0answers
44 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
votes
1answer
49 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
votes
0answers
44 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
votes
1answer
30 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 ...
4
votes
0answers
49 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
votes
1answer
49 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
votes
1answer
38 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
56 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 ...
1
vote
1answer
71 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
33 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 $A^\dagger$ ...
1
vote
0answers
75 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 ...
3
votes
1answer
45 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
54 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 ...
1
vote
0answers
32 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
vote
1answer
37 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
votes
0answers
17 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
vote
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
vote
1answer
28 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 ...
1
vote
1answer
61 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
votes
1answer
54 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.