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

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Uncountable Kronecker Delta?

If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice). Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set ...
2
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2answers
21 views

not understanding a step in a proof

Hi: I'm reading some introductory notes on hilbert spaces and there is a step in a proof that I don't follow. I will put the exact statement below. If someone could explain how it is obtained, it's ...
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0answers
22 views

Normal Operators: Superalgebra (II)

Problem highlighted at the end! Application Reduction to only one operator!! Reference This builds up on: Superalgebra (I) Convention All operators possibly unbounded!! Structures Given a ...
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2answers
70 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
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0answers
13 views

Differential second in infinite dimension

Let $f: E \to R$ sending $x \mapsto \|x\|$ and make some simple hypothesis: with $E$ an Hilbert Space Let say that the norm $\|\bullet\|$ is derived from a scalar product So we can easily find ...
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22 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
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1answer
29 views

Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
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0answers
19 views

Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
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1answer
23 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
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1answer
91 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 ...
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1answer
32 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha:\mathcal{D}H_\alpha\subseteq\mathcal{H}_\alpha\to\mathcal{H}_\alpha:\quad H_\alpha=H_\alpha^*$$ And their ...
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3answers
59 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 ...
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2answers
102 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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1answer
62 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: ...
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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
33 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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4answers
460 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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0answers
24 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. ...
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2answers
37 views

Ordering : Ranges

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Note that one has: $$\Delta ...
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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 $\Omega_1 \cap \Omega_2=\emptyset$. ...
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1answer
26 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 ...
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1answer
14 views

Continuous function space and Reproducing kernel Hilbert [closed]

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 ...
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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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1answer
41 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 ...
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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 ...
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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
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0answers
37 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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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: ...
0
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0answers
40 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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57 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 ...
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1answer
25 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
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2answers
78 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. ...
2
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1answer
58 views

Proving that an orthonormal system close to a basis is also a basis

Let $\mathcal{H}$ be a Hilbert space and $(e_n)_{n \in \mathbb{N}} \subseteq\mathcal{H}$ be an orthonormal basis and $f_n$ be an orthonormal system such that $(f_n)_{n \in \mathbb{N}} ...
4
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0answers
30 views

A question about equivalence of norms involving infimum

Let $I$ be a Banach space with norm $\lVert\cdot\rVert_I$. The norm $$\inf\{\lVert(G_i(u_i))_i\rVert_{\ell^2}\mid u=\sum_{I \geq 0}u_i\}\qquad\text{is equivalent to}\qquad \lVert{u}\rVert_{I}$$ where ...
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1answer
26 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: ...
2
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1answer
40 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: ...
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1answer
21 views

Normal Operators: Examples

Given the Hilbert space $\mathbb{C}^2$. Consider bounded opertors: $$N:\mathbb{C}^2\to\mathbb{C}^2:\quad\|N\|<\infty$$ Then there are some with: $$N\neq N^*\quad N^*N=NN^*$$ What examples are ...
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1answer
73 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
2
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1answer
23 views

Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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1answer
26 views

Show that a subspace is closed in a Hilbert space $H$

If $T$ is a bounded linear operator in a Hilbert space $H$, and $T$ is self-adjoint and is equal to its inverse, how can I show that $\widehat{H} = \{h + Th : h \in H\}$ is closed? If I consider the ...
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1answer
14 views

Normal operators on a hilbert space over the reals - does $norm(Tx)=norm(T^*x)$ imply $T$ normal?

The title states the question. It's easy to prove the result for scalars C via polarisation identities but I don't think the same method works in the real case: Let $S=TT^*-T^*T$ then one obtains ...
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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 ...
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1answer
40 views

$h_1,\ldots,h_n$ are linearly independent if and only if $A$ is invertible

Show that $h_1,\ldots,h_n$ are linearly independent in $\mathscr{H}$ (Hilbert space) if and only if the matrix $A$, defined by $A=[a_{ij}]$ where $a_{ij}=\langle h_j, h_i \rangle$, is invertible. ...
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1answer
29 views

Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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2answers
44 views

Dense Operators: Kernel

This thread is Q&A. Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a dense operator: ...
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1answer
37 views

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$ for all $n\in \mathbb{N}$. Let ${e_n}$ be a ...
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1answer
22 views

Wave Operators: Isometry

This thread is only Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
32 views

Isometries: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Consider isometries: $$R_\lambda\in\mathcal{B}(\mathcal{H}):\quad R_\lambda^*R_\lambda=1$$ Then it follows: $$R_\lambda\rightharpoonup R\implies R_\lambda\to R$$ ...
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0answers
101 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: ...
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1answer
30 views

Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...