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

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7
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2answers
71 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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vote
0answers
25 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
2
votes
0answers
80 views
+100

Polar Decomposition: Adjoint

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\subseteq\mathcal{H}\to\mathcal{K}:\quad A=A^{**}$$ And its decompositions: ...
0
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0answers
13 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha=H_\alpha^*:\quad U_\alpha(t):=e^{-itH_\alpha}$$ Then for their sum: $$H=\bigoplus_\alpha H_\alpha\implies ...
1
vote
1answer
84 views

Spectral Measures: Square Root

Isometric Equality Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$ Denote for shorthand: $$H:=A^*A:\quad H=H^*$$ Regard elements: ...
2
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1answer
52 views

Polar Decomposition: Unitarity

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
3
votes
3answers
47 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
0
votes
1answer
59 views

Trace: Independence

Problem Given a Hilbert space $\mathcal{H}$. Consider an operator: $$A\in\mathcal{B}(\mathcal{H}):\quad\operatorname{Tr}|A|<\infty$$ Regard ONB's: ...
1
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1answer
42 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 ...
0
votes
1answer
32 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-Schwarz: ...
1
vote
1answer
67 views

Spectral Measures: Embedding

This thread is just a note! Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
9
votes
4answers
439 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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1answer
89 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
1
vote
1answer
18 views

Are they true these generalizations from matrices to operators about functional calculus?

Motivation: If we have some real function $f$ defined on an interval $I$ and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ is a diagonal matrix such that $\lambda_i \in I$ for all $1\leqslant i ...
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0answers
32 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
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1answer
15 views

The norm of a bounded linear functional on a Hilbert space is the norm of the vector?

If $L$ is a bounded linear functional on a Hilbert space $H$, then we know that $$Lx=(x,y),\quad \forall x\in H,$$ for some $y\in H$. Is it true that $\|L\|=\|y\|$? We have by Cauchy-Schwarz that ...
0
votes
2answers
30 views

Ordering : Ranges

Given a Hilbert space $\mathcal{H}$. Consider operators: $$A,A'\in\mathcal{B}(\mathcal{H}):\quad A=A^*\quad A'=A'^*$$ Introduce an ordering: $$A\leq A':\iff\sigma(A'-A)\geq0$$ Then one has: $$0\leq ...
0
votes
1answer
38 views

Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$

Suppose $M$ is a closed subspace of a Hilbert space $X$. Let $x\in X$. Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$ My Try: First of all I am confused ...
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0answers
23 views

Conditions for an operator on a Hilbert space to have an orthonormal set of eigenfunctions

I'm working on a problem that requires the following operator, $A^TA$, to have an orthonormal set of eigenfunctions. Note $A:H_1 \mapsto H_2$, where $H_1$ and $H_2$ are separable Hilbert spaces. ...
0
votes
1answer
46 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
1answer
56 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
1
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1answer
66 views

Norm of two operators

(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$ For this one I tried the ...
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1answer
24 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
0
votes
1answer
40 views

$L^2$ inner product split over sub-domains in $\mathbb{R}^3$

I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. ...
0
votes
1answer
24 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
3
votes
1answer
361 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 ...
4
votes
1answer
760 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 ...
0
votes
1answer
14 views

Continuous function space and Reproducing kernel Hilbert [on hold]

Let $E=C[-1,1]$, space of all real-valued continuous functions on [-1,1], $E$ is a reproducing kernel hilbert space? by inner product $\int_{-1}^{1} f(x)g(x) w(x) dx$ where $w(x)>0$ is weighted ...
1
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1answer
34 views

What is $\mathrm{dim}(\mathrm{Sym}(\mathrm{Herm}(H)^{\otimes N})$?

The totally symmetric subspace of $(H^k)^{\otimes N}$, with $H^K$ a $k$-dimensional Hilbert space, has dimension $\binom{N+k-1}{k-1}$. But I now want to know the dimension of the totally symmetric ...
2
votes
1answer
39 views

Normal Operators: Von-Neumann

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}N\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Regard their algebra: ...
0
votes
1answer
15 views

Elementary inequality with norms

Is it reasonable to hope the validity of an inequality of this form, where $H$ can be any separable Hilbert space $$\forall \alpha \geq 0 \, \exists \gamma \geq 0: || x - \alpha y|| \leq \gamma || x ...
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0answers
33 views

$\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2$

I want to show that $\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2((0,1); \mathbb{C}) =: X$. Of course the only problem is to show completeness. In our lecture we ...
2
votes
2answers
79 views

Spectral Measures: Concentration

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$ Define its support: ...
0
votes
1answer
59 views

Projections: Beppo Levi

Given a Hilbert space $\mathcal{H}$. Consider projections: $$P_\lambda\in\mathcal{B}(\mathcal{H}):\quad P_\lambda^2=P_\lambda=P_\lambda^*$$ And directed indices: ...
1
vote
2answers
56 views

Why is the sequence $(\langle x_n,a \rangle)$ Cauchy when $(x_n)$ is?

Let $\mathcal H$ a Hilbert space over $\mathbb R$ and $A = \{x\in \mathcal H : \langle x, a \rangle \geq 1 \}$. I'm trying to prove that $A$ is closed. Let $(x_n) \subset A$ be a Cauchy-sequence. ...
4
votes
2answers
69 views

Is a bounded operator with finite trace trace class?

Let $\mathcal{H}$ be a seperable 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 ...
2
votes
2answers
62 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...
0
votes
2answers
82 views

Sesquilinear Forms: Parallelogram

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose one has: ...
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0answers
44 views

Hermitian conjugate of an antiunitary operator

In certain fields of quantum mechanics, one often considers symmetry transformations which are defined in terms of operators which do not change the norm of states in the Hilbert space. For the ...
2
votes
1answer
65 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
0
votes
1answer
24 views

Common solutions to quadratic equations associated to self-adjoint matrices

Let $\mathcal{H}$ be a complex Hilbert space of dimension $d<+\infty$, and let $\{|n\rangle\}$ with $n=0,\cdots,d$ be an orthonormal basis in $\mathcal{H}$. Let $\mathbf{A}$ be a self-adjoint ...
2
votes
0answers
32 views

Gradient of inner product in Hilbert space

Let $\mathcal{H}$ be a Hilbert space and \begin{align} f&\colon \mathcal{H} \to \mathbb{R}\\ f(x) &= ||x-c||_\mathcal{H} ^2 \end{align} from some constant $c \in \mathbb{H}$ Is the derivative ...
0
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0answers
39 views

Countably Infinitely Many Points in a Euclidean Space

Do there exist $d\in\mathbb{N}$ such that there are pairwise distinct points $x_1$, $y_1$, $x_2$, $y_2$, $\ldots$ in $\mathbb{R}^d$ such that (i) $\left\|x_i-y_i\right\|_2 >1$ for ...
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0answers
35 views

Normal Operators: Superalgebra

Problem Given a Hilbert space $\mathcal{H}$. Consider dense operators: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{H}:\quad\overline{\mathcal{D}A}=\mathcal{H}$$ Denote the normals: ...
0
votes
1answer
21 views

Spectral Measures: Pushforward

This thread is Q&A. Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
votes
2answers
59 views

Spectral Measures: Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
0
votes
1answer
52 views

Domain Issue: Notation

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ It is well known that:* $$A=A^{**}\iff ...
3
votes
1answer
24 views

Bessel potential space: Proof of completeness

I want to know a proof that the (one-dimensional) Bessel potential space (for $p=2$) $$H^s(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}:\int_{\mathbb{R}}(1+\lvert \xi\rvert^2)^{\frac{s}{2}}\lvert ...
2
votes
2answers
77 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
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1answer
450 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: $M$ is a non-empty set in an Inner Product Space (IPS) $X$. I need to show that the annihilator of $M$ which is ...