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

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Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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1answer
63 views

If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
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29 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
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39 views
+100

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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88 views

Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
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2answers
373 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
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help Hilbert space [on hold]

Let $H$ be a Hilbert space, $T:H \longrightarrow H$ a linear bounded operator, and $||T||$ the norm of $T$. Help me to prove that $||T||=\sup \{ ||T(x)|| : ||x|| \le 1 \}$.
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Terminology for orthogonal projections

Let $H = X \oplus Y$ a Hilbert space. Then, the map $p(x + y) = x$ is called the orthogonal projection onto $X$ along $Y$. Why is it necessary to mention along $Y$? Of course if a space has a ...
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1answer
26 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
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1answer
89 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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21 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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1answer
32 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
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19 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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1answer
34 views

Fréchet derivatives of $\sum_{n=1}^\infty x_n^2/n^3 -\sum_{n=1}^\infty x_n^4$

I read that the second order Fréchet derivative $F''(0)$ of linear functional $F:\ell_2\to\ell_2$, where $\ell_2$ is the separable real Hilbert space, defined by ...
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1answer
38 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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85 views

If every $M\subset X$ closed is such that $M^{\bot\bot}=M$, then $X$ is Hilbert space. [closed]

If $X$ is an inner product space and if $M^{\bot\bot}=M$ for every closed subspace $M$ of $X$, then $X$ is a Hilbert Space. Can someone help-me?
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22 views

Absolute continuity and Hilbert Space [closed]

$$H=\{f(x)\,\,\Big|\,\, f(x),f'(x),f''(x) \text{ are absolutely continuous; }f'''(x)\text{ is }L_2\text{ on }[-1,1]\}$$ with the inner product as $$\langle f,g\rangle ...
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1answer
14 views

Show that this operator is linear

Let $\Bbb H$ is a Hilbetr space and $T:\Bbb H\to\Bbb H$ be a operator such that $$<x,Ty>=<Tx,y>$$ $\forall x,y\in\Bbb H.$ I want to show that $T$ is linear and bounded. If I can show that ...
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29 views

Basis of $W^{1,p}_0\cap L^2$ using $(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}$

Let $p > 1$. Define $\lambda_i$ by the eigenfunctions of the problem $$(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}\quad\text{for all $v \in H^s_0(\Omega)$},$$ where $s$ is chosen ...
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35 views

Derivative of norm in Hilbert space

I read (p. 485 here) that the Fréchet derivative of norm (non-linear) functional $p:H\to\mathbb{R}$, $x\mapsto\|x\|$ is $\frac{x}{\|x\|}$ for all $x\ne 0$, which I think to be intended as the linear ...
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1answer
29 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
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46 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
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23 views

Sufficient conditions for weak continuity

Are there any "easily verifiable" sufficient conditions for weak (equivalently, weak*) continuity of (not necessailry linear) maps on the unit ball of $\ell^2$, mapping into $\ell^2$? Apologies for ...
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1answer
16 views

Subspace of a Hilbert space with a distinct inner product

I don't really know where to begin with the following question: Let $ (H_0, \langle \cdot \rangle_0)$ be a closed subspace of $ (H, \langle \cdot \rangle )$ such that norms induced by $ \langle \cdot ...
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1answer
20 views

Proving an orthogonal projection of the Hilbert adjoint is just the adjoint

I'm facing the following problem: let $ H_0 \subset H $ be a $ T$-invariant closed subspace of Hilbert space $ H $ (i.e. $ T(H_0) \subset H_0 $) and $ P$ - an orthogonal projection of $ H $ onto $ ...
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1answer
15 views

If $E$, $\overline{E}$ are orthogonal projections such that $\mathrm{range}(\overline{E})=\overline{\mathrm{range}(E)}$, then is $E\ge\overline{E}$?

I feel like this should be true. Let $\mathrm{range}(E)=A$ and $u$ be an arbitrary vector in a Hilbert space $H$, it is sufficient to show $\langle (E-\overline{E})u,u\rangle=0$. By Cauchy-Schwartz: ...
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1answer
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About the von Neumann decomposition

The von Neumann theorem states that for any symmetric operator $f$, the domain $D_{f^\dagger}$ of its adjoint $f^\dagger$ is the direct sum of the three subspaces $D_{\bar{f}}$, $\aleph_z$, and ...
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1answer
21 views

CAR-Algebra: Nontriviality?

Given a Hilbert space $\mathcal{h}$. Consider the abstract CAR-algebra $a:\mathcal{h}\to\mathcal{A}_\text{CAR}$. Then their actually isometries: $$a:=a(f):\quad ...
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27 views

Any example of non-closed operator?

I cannot think of one. By the way, is there any good exercise book on functional analysis or hilbert space?
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continuos spectrum of $R+L$, where $R$ and $L$ are the right and left shift of sequences in $l_2$

consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I'm trying ...
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1answer
38 views

Unit sphere weakly dense in unit ball

I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball. I already had to prove that the unit ball ...
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1answer
40 views

Prove the operator on hilbert space is compact

My question is actually the same as the first part of this one, Prove that T is compact which has not been answered. I am thinking about two ways, 1) use a bounded sequence $\{g_n\}$, and try to ...
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1answer
49 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
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28 views

Why cannot a densely defined operator be extended to an everywhere defined operator?

I am a physicist learning functional analysis because of its fundamental role in quantum mechanics. There are so many bizarre facts. One is, there are densely defined operators which seem cannot be ...
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39 views

Are all Banach spaces also Hilbert spaces?

We have the well-known "polarization identity" $$(x,y)=\frac{1}{4}\left(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2\right)\tag{1}$$ that works in any Hilbert space. Hence, is every Banach space also a ...
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15 views

Why only densely defined operators can have an adjoint operator?

Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?
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123 views

An example of non-closed subspace of a Hilbert space?

I am reading a book on Hilbert space. It seems that the author assumes that a linear subspace of a Hilbert space can be non-closed. I cannot think of an example. I am still used to the ...
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Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
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1answer
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0 limit point of spectrum of completely continuous operator $H\to H$

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 475 here) that 0 is an accumulation point for the spectrum of a completely continuous operator $A:H\to H$ where $A$ ...
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117 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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1answer
24 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
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17 views

Orthonormal Hamel Basis is equivalent to finite dimension

Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a ...
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Let $H$ be a Hilbert space, $A$ is unitary and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$?

Let $H$ be a Hilbert space, and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$? What I have is if $x\in S^{\perp}$ then $x\perp A(A^*A^*Ax)$ then $(x,A^*Ax)=(Ax,Ax)=0$, so $x$ is in ...
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On existence of an element whose is orthogonal with given $n$ elements of a Hibert space of infinite dimension

Let $H$ be a Hilbert space of infinite dimension with the scalar product $\left\langle {.,.} \right\rangle $. Given $u_1,...,u_n\in H$. Is there a $u\in H$, $u\ne 0$ for which $\left\langle {u,{u_j}} ...
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1answer
71 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
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the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
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26 views

Calculation of operator norm

$H$ is a Hilbert space, $T: H \to H$ linear bounded operator, $||T||$ is the norm of $T$ given by $$||T||=\sup\{||T(x)||;||x||\le 1 \}. $$ Is it true that $$||T||=\sup\{|\langle ...
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1answer
27 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
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1answer
30 views

Direct sum of kernel and image of the adjoint operator

Let $H$ be a separable Hilbert space and $T:=I-A$, where $A:H\to H$ is a compact operator. If $T^\ast$ is the adjoint operator of $T$ it can be proved that $\ker T$ and $\text{im } T^\ast:=T^\ast (H)$ ...
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28 views

Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...