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

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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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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]}...
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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 ,...
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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} ...
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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 ...
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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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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 ...
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1answer
49 views

Prove a sequence is bounded under a Hilbert space

Let $T:H\to H$ be defined by $Tx=\sum_{n=1}^\infty \lambda_n \langle x,\varphi_n \rangle \varphi_n$ where $\{\varphi_n\}_{n=1}^\infty$ is an orthogonal sequence and $\{\lambda_n\}_{n=1}^\infty$ is a ...
2
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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 ...
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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] =...
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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 $u$ ...
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1answer
56 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 $U\subseteq V$ ...
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0answers
49 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 $...
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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: ...
2
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1answer
35 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 $X$, ...
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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 ...
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1answer
20 views

Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
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21 views

Fourier basis of $L^2([-\pi,\pi])$

I have read that the Hilbert space $L^2([-\pi,\pi])$ has a Hilbert basis: $$\{e^{inx}|n\in\Bbb{Z}\}$$ This to me indicates that we can only represent a function $u(x)$ by a Fourier Series iff $u(x)\in ...
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0answers
16 views

integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
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1answer
15 views

Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
2
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1answer
32 views

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Proof of lemma $9.7$ in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations argues as follows: For an element $u \in H_0^1(\Omega)$ we define $D_h u= \frac{u(x+h)-u(x)}...
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0answers
18 views

Fatou lemma and weak convergence in Hilbert

In a Hilbert space $H$ a sequence $(x_n)_{n\geq0}$ is said to converge weakly to $x$ if $\forall y\in H:\langle y,x_n\rangle\rightarrow\langle y,x\rangle$, the case in which we can easily deduce an ...
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2answers
149 views

Is a bounded operator with finite trace trace class?

Let $\mathcal{H}$ be a separable Hilbert space, $A\in\mathcal{B}(\mathcal{H})$ a bounded linear Operator and assume we have an orthonormal basis $(x_n)_{n=1}^\infty$. If $A$ is trace-class, then $\...
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0answers
31 views

Distance preserving function on hilbert space

It is known that an isometry on B(H) is distance preserving .I am trying to show ,conversely , that if F=R,every distance preserving function f on H( Hilbert space) has the form f(x) = f(0) +Tx for ...
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1answer
20 views

Conjugate linear isometry of Hilbert operators

Let $H$ and $R$ be Hilbert spaces and consider an operator $T$ in $B(H,R)$. I need to show that there is a unique operator $T^*$ in $B(R,H)$ satisfying $$(Tx│y)_R = (x│ T^* y)_H,$$ $x \in H$, $y \in R$...
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1answer
36 views

Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
2
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1answer
20 views

Orthogonal sequences under A Hillbert space

I know that for two vectors $u,v\in H$ where $H$ is a Hilbert space the definition for orthogonality is $\langle u,v \rangle =0$. is thaat also corret for sequences? What is the definition for ...
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0answers
46 views

Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$ T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy $$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...
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1answer
31 views

Linear Operator on Hilbert Space $l(\mathbb Z)$

Let $A$ be the linear operator on $l(\mathbb Z)$ defined for $u=\{u_k\}_{k \in \mathbb Z}$ as $(Au)_k = \sum_{h=-\infty}^{+\infty}a_{k,h}u_h$ where $a_{k,h}=\frac{1}{(k-h)2}$ for $h \not= k$, and $...
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0answers
25 views

Minimal uniquely achieved of a funcional in space Hilbert

I have the follow problem : Let be $H$ a Hilbert space and $E \subset H$ a closed space of $H$. Give a $w \in H$ such that $w \not \in E^\perp$, show that $$\inf_{v \in E, |v| = 1}\big<w,v\big> ...
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1answer
36 views

$L^2$ convergence of partial sum of a sequence of functions: $\sum_{k=1}^n\frac{g^k(x)}{k}$

Let $g:\mathbb{R}\to \mathbb{C}$ be an $L^2$ function such that $|g(x)|\leq\epsilon<1, $ for every $x\in\mathbb{R}$, and $g(x) = O\left(\frac{1}{x}\right)$. I want to know if $h_n(x):= \sum_{k=1}^n ...
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1answer
33 views

Can we find a concrete representation of $\iota\iota^\ast y$, if $\iota$ is a Hilbert-Schmidt embedding between Hilbert spaces?

Let $U$ and $H$ be real Hilbert spaces $\iota:U\to H$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ Can we find a concrete representation of $Qy$ for some $y\in H$? By Riesz' ...
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36 views

Square-root of $\iota\iota^\ast$, where $\iota$ is an isometric embedding between Hilbert spaces

Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)...
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130 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall x \...
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0answers
30 views

Given a special Hilbert space $U_0$, is there a proper superspace $V$ such that the inclusion $\iota:U_0\to V$ is Hilbert-Schmidt?

Let $U$ be a Hilbert space $Q\in\mathfrak L(U)$ 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\rangle_U\;\;\;\text{for }u,v\in ...
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1answer
24 views

The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
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1answer
33 views

If a positive operator $y$ has the same kernel as $cy$, what can we conclude about the kernel of $c$?

Let us consider the equation $x=cy$ in $B(H)$. Assume that: $y$ is a positive operator. $x$ and $y$ have the same null space. Ker($y$) is contained in Ker($c$). Can we conclude that Ker($y$)=...
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2answers
51 views

Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp $

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...
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1answer
44 views

Show that a sequence does not converge

I have a very similar question to this and so I'll change my letters to match: Let $\{\varphi_n\}^∞_{n=1}$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{λ_n\}^\...
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3answers
164 views

Convergence under a Hilbert space

Let $\{\varphi_n\}_{n=1}^\infty$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{\lambda_n\}_{n=1}^\infty$ be a sequence of numbers Define $T:H\to H$ by $Tx= \sum_{n=...
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0answers
26 views

Proof Verification: Separable Hilbert Space has a Countable Orthonormal Basis?

I was browsing proofs of this involving the Gram-Schmidt process when the following occurred to me. I'd appreciate feedback. If $H$ is a separable Hilbert space then it has a countable dense subset $...
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0answers
18 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be 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\rangle_U\;\;\;\text{for }...
4
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2answers
528 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As $H^*...
4
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1answer
33 views

Is the Null Space of an linear operator the same with the Null Space of its associated hermitian?

Let A be a bounded linear operator on $H$ where $H$ is a (not necessary I think, but in my case separable) Hilbert space. Then, the question: is its null space the same as the null space of the ...
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1answer
48 views

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Equip $C_0^\infty(\mathbb{R}^n)$ with the norm $$ \|\varphi\|^2_1 := \int_{\mathbb{R}^n}| \nabla \varphi|^2dx.$$ Indeed, $\| \cdot \|_1$ is a norm on $C_0^\infty(\mathbb{R}^n)$ because any constant ...
3
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2answers
96 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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2answers
27 views

For a densely defined symmetric operator $A$, is $A^2$ also densely defined?

Let $A : D(A) \to H$ be a possibly unbounded, densely defined symmetric operator on a Hilbert space $H$ ($A$ being symmetric means that $(\varphi, A\psi) = (A\varphi, \psi)$ for all $\varphi, \psi \in ...
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3answers
1k views

Orthonormal basis in $L^2(\Omega)$ for bounded $\Omega$

In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find a orthonormal basis $\{e_n\}_{n\in {\bf Z}}$ for $L^2(\Omega)$ where $$e_n(x)...
0
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0answers
36 views

Norm of an operator in a Hilbert space

Let $T\neq 0, \neq I$ be a linear operator of a Hilbert space such that $T \circ T = T $. Show that $\|T\|=\|I-T\|$. Anyone has an idea ? I just proved that $\|I-T\| \leq 1 + \|T\|$ but it is not ...
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1answer
41 views

A factorization for operators

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$. Q) Can we ...