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

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2answers
30 views

The form of a normal operator with only one element in its spectrum

Let be $H$ a Hilbert space. Show that if $T$ is a normal linear operator continuous (i.e. $T^*T = TT^*$, with $T^*$ the Hilbert adjunct of $T$) and your spectrum $\sigma(T) = \{\lambda\}$, than $T = \...
0
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0answers
14 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
2
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0answers
31 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...
-1
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0answers
35 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
0
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1answer
34 views

For two positive operators on Hilbert space is it true that $A \ge B \implies \|A\| \ge \|B\|$?

$H$ is Hilbert space. $A$ and $B$ is positive linear operators from $H$ to $H$ i.e. $\forall x\in H\, (Ax,x),\,(Bx,x)\ge 0$. $A\ge B$ means that $A-B$ is positive. Does that means that $\|A\| \ge \|B\|...
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2answers
31 views

Question on operator of hilbert space, why $f(x)=\sum_{i}(f|e_i)e_i$?

let $(V,(. |.))$ a Hilbert space. Let $\{e_i\}_{i=1}^\infty $ an orthonormal basis and $f:V\to V$ a linear application. Here are my questions : 1) Why $f(x)=\sum_{i=1}^\infty (f,e_i)e_i$ ? 2) Why ...
0
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2answers
47 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
2
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1answer
35 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
0
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0answers
33 views

If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
3
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2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
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0answers
25 views

Extending a unitary operator

Suppose that $\mathcal{H}_1$ and $\mathcal{H}_2$ are two separable Hilbert spaces and that $X\subset \mathcal{H}_1$ is a dense subspace (i.e. $\overline{X}=\mathcal{H}_1$). If $\operatorname{W}:X \to \...
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2answers
74 views

Quantum Mechanics: position and the separability of Hilbert space?

I would be pleased if someone could point out to me where I go wrong in the following sequence of statements: One model of quantum mechanics identifies states of a particle with normalized vectors (...
2
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0answers
52 views
+50

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
1
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0answers
33 views

How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
0
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1answer
46 views

Prove that $\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$. [closed]

Let $T$ be a self adjoint bounded linear operator in a Hilbert space $H$. Prove that $$\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$$
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1answer
24 views

Property of inner product on Hilbert space.

Let $H$ be a Hilbert space equipped with inner product $\left< \cdot , \cdot \right>$. Fix $u\in H$ and constant $R_0 > 0$. Define subset $K_u(R_0)$ of $H$ by $$K_u(R_0) =\{w \in H : \left&...
1
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1answer
37 views

What does the weak* topology on $\ell_2$ look like?

I am wondering about a way to construct a base or subbase for the weak* topology on $\ell_2$. I am fairly new to topology and functional analysis, so I apologize if the question is not precisely ...
3
votes
3answers
51 views

Existence of rotations between two points

Let $x,y\in\mathbb R^n$ ($n\in\mathbb N$) be two given points with the same Euclidean norm: $\|x\|=\|y\|$. Does there, in this case, exist an orthogonal matrix $U\in\mathbb R^{n\times n}$ such that $$...
2
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0answers
32 views

Sobolev Space with partial inner product

In my work, I encountered the following problem. Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \...
2
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0answers
43 views

Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
0
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0answers
58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
0
votes
1answer
25 views

Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
2
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0answers
39 views

Example of Hilbert space non isomorphic to $L2$

I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of $L^{2}(X,\mu)$ spaces nor subespaces of $L^{2}(X,\mu)$. Some idea to start? Thanks everyone.
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1answer
17 views

Sesquilinear forms - How does positiveness imply hermitianity?

In my mathematical methods for physics course notes I find this: A positive sesquilinear form is nondegenerate and Hermitian The first statement is trivial: a ...
0
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0answers
22 views

weighted shift operator for complex Hilbert space

I am trying to solve that if H is a complex Hilbert space with orthonormal basis $\{e_n\}_{n=1}^{\infty}$ and let $\{a_n\}_{n=1}^{\infty}$ be a sequence with $\lim_{n\rightarrow}a_n = 0$. Define the ...
2
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0answers
31 views

Completeness of 'Hardy Space' $H^2(D)$

Define Hardy Space $H^2(D)$ as a space of holomorphic functions $f$ on unit open disc $D=\{z\in\mathbb{C}:|z|<1\}$ endowed with the norm $$ ||f||^2=\sup_{0<r<1} \int_0^{2\pi} |f(re^{i\...
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0answers
63 views
+50

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
0
votes
1answer
34 views

Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
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0answers
13 views

Application of Uniform boundedness theorem: $\langle Tx,y\rangle$ bounded for each $x,y$ then $||T||$ is bounded

For Hilbert Space $X$, if we have a condition on a subset $F\subset B(X)$ ('set of bounded linear operators on $X$') such that $$ \{\langle Tx,y\rangle:T\in F\} $$ is a bounded set for each $x,y\in ...
1
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1answer
23 views

Characterization of orthogonal projections in terms of operator norms

I want to show the following equivalence: If $X$ is a Hilbert Space and $P\in B(X)$ (i.e. $P$ is bounded and linear) and $P^2=P$, then $$ (\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1 $$ I know that ...
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0answers
30 views

Hilbert Spaces and Hamel Basis

Let $H$ be a Hilbert Space of infinite dimension, $S$ a not finite orthonormal basis and $B$ a Hamel basis to $H$. i) How to show that the cardinality of $B$ is greater than or equal to the ...
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0answers
20 views

G. Vitali's Result

Let $(x_n)_{n\in\Bbb N}\subseteq\mathcal L_2([a,b])$ be an orthonormal sequence. I want to prove the following: $(x_n)_{n\in\Bbb N}$ is complete $\Leftrightarrow\sum_{n=1}^\infty \big|\int_{[a,t]}...
2
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1answer
22 views

Stuck on elementary proof on completeness of $W^{1,2}(\mathbb{R})$ as Hilbert Space

Let $W^{1,2}(\mathbb{R}):=X$ be the space of continuous functions $f$ such that $f\in L^2(\mathbb{R})$ and there exists $f'\in L^2(\mathbb{R})$ such that $$ f(b)-f(a)=\int_a^b f'(t)\,dt $$ for ...
1
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1answer
37 views

Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
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0answers
23 views

Hilbert space mean ergodic theorem application

Let $(u_n)_{n \geq 0}$ be a bounded sequence in a Hilbert space. We define $$ s_h = \limsup \frac 1 N \sum_{n=o}^{N-1} \langle u_{n+h} , u_n \rangle $$ Show that, if $ \lim \frac 1H \sum_{h=o}^{H-1} ...
2
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1answer
49 views

Two orthonormal sets in a Hilbert space. One is complete, the other must be complete.

Given two orthonormal sets $\{e_k\}_{k=1,2\ldots}$, $\{e'_k\}_{k=1,2\ldots}$ in a Hilbert space $H$, which satisfy $$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*} $$ Prove that if $\{e_k\}_{k=1,2\...
1
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1answer
22 views

How does the usual properties of Hilbert adjoint operator follow from this definition?

Given two hilbert spaces $X,Y$, and a bounded linear $T:X\to Y$, define $S:Y\to X$ by $$ S=J_{X}^{-1} \circ T' \circ J_Y $$ Where $T':Y'\to X'$ is given by $T'(y')=y'\circ T$ for $y'\in Y'$ and $J_X ...
0
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0answers
26 views

If $Z$ is a closed subset of Hilbert Space $X$, is it true that $Z\neq X \implies Z^{\perp}\neq \{0\}$?

It is clear from Projection theorem that if $Z$ is a subspace, then since $X=Z\oplus Z^{\perp}$, $Z^{\perp}$ is not trivial (By the way, is there any reasoning that would show this without referring ...
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2answers
64 views

Eigenspace and eigenvector inside a Hilbert space

Given $\{v_n\}_{n=1}^\infty$ an orthonormal sequence in a Hilbert space. Let $\{\lambda_n \}_{n=1}^\infty$ a sequence of numbers and $F:H \to H$ defined by $Fx=\sum_{n=1}^\infty \lambda_n \langle x ,...
0
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1answer
46 views

How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$ H^k = W^{k,2}. $$ I've also seen the following exercise recently: $$ \frac{1}{2}u'' = 1 $$ And here I'm supposed to find out if $u$ ...
2
votes
1answer
53 views

How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
0
votes
1answer
55 views

How can we compute the adjoint of the inclusion between two Hilbert spaces?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ ...
2
votes
1answer
54 views

Find spectrum of integral operator

Let Af(x) = $\int_0^1 K(x,y)f(y)dy$, $A:L_2[0,1]\rightarrow L_2[0,1].$ Where $K(x,y) = \sinh(\min(x,y)\sinh(1-\max(x,y)). $ where $\sinh(x) = \frac{e^x - e^{-x}}{2}$ Find $\sigma(A), ||A||.$ I ...
1
vote
0answers
48 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
0
votes
1answer
25 views

Riemann-Lebesgue Lemma (general)

Let $\big(X,\langle\ \rangle\big)$ be a Hilbert space over $K$. I want to prove the following If $(x_n)_{n\in\Bbb N}$ is an orthonormal sequence in $X$ $\Rightarrow\; x_n\to0$ weakly My attempt: ...
0
votes
1answer
29 views

Inversible operator in Hilbert space

Consider $\phi\in L^{\infty}[0, 2\pi]$. Let M be operator $L_2[0, 2\pi]\rightarrow L_2[0, 2\pi]$$$Mf = \phi f$$ In $L_2[0, 2\pi]$ we have topological basis ${e^{inx}}, n\in \mathbb Z$. $L_2[0, 2\pi] =...
2
votes
1answer
34 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
0
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0answers
55 views

Generalized Absolute Value II

Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...
0
votes
1answer
22 views

The meaning of an orthogonal basis?

I am reading up on Hilbert spaces and am a bit confused about the properties of an orthogonal basis. Would I be correct in saying that we can define an orthogonal basis as: Every element in the ...