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

learn more… | top users | synonyms

1
vote
0answers
48 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
0
votes
1answer
25 views

Bijective bounded linear operator is invertible

The following is an exercise from Halmos book "A Hilbert space problem book" : Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and ...
0
votes
2answers
41 views

$\int_{\mathbb R}|f(x)|^{2} dx <\infty \implies \sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx <\infty$?

Let $f\in L^{2}(\mathbb R),$ that is, $\int_{\mathbb R}|f(x)|^{2} dx <\infty,$ and $\beta>0.$ My Question: Is it true that that: $\sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx ...
0
votes
0answers
31 views

Differential of a mapping $F:X\rightarrow X$

I'm reading a paper and says: The differential of a mapping $F:X\rightarrow X$ with respect to $v$ is: $D_vF(x)\in{\cal L}(X)$ where $H$ is a Hilbert space and ${\cal L}(X)$ stands for ...
0
votes
1answer
22 views

Z-transform and$ H_2$ space

The following is from the preliminaries of a paper. Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function $G:(\mathbb{C} \cup \{\infty\})\backslash ...
2
votes
1answer
60 views

Dense subset of Hilbert space has trivial orthogonal complement

If $D$ is a dense subset (not subspace) of a Hilbert space then is the orthogonal complement of $D$ equal to $\{ 0 \}$? This is true if $D$ is a subspace but if you only know that $D$ is a subset ...
2
votes
0answers
38 views

A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
1
vote
1answer
50 views

Convergence of a series

Can someone help me how to do the following? Let $H$ be a Hilbert space with inner product $\langle \cdot,\cdot \rangle$, and orthonormal basis $(e_n)$. For $x,y \in H$ prove that the series in the ...
1
vote
1answer
32 views

Show that in a complex Hilbert space, T normal bounded linear operator, $\| T^2 \| =\| T \| ^2$

So, as a part of a problem, I've been asked to prove that if $H$ is a complex Hilbert space and $T\in L(H)$ is normal, then $\| T^2 \| =\| T \| ^2$ (Operator norm) Context: This is part (b) in a ...
0
votes
1answer
44 views

Resolvent: Decay Behavior [closed]

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the resolvent set: ...
1
vote
0answers
47 views

Every Hilbert space is connected

Let $H$ be a Hilbert space. Proving $H$ is connected, suppose $\{e_i\}_{i\in I}$ is a orthogonal basis of $H$. Thus $H=\bigoplus_{i\in I} \Bbb C e_i$. Clearly $\Bbb Ce_i$ is connected for every $i$, ...
1
vote
1answer
28 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
1
vote
1answer
40 views

Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
2
votes
1answer
62 views

Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
0
votes
1answer
24 views

Question about ortogonality on $L^2(\Omega)$

Let $u\in L^2(\Omega)$. Is the following proposition true? $\big(\forall v\in H^1_0(\Omega)\big)\quad (u,v)_{0,\Omega}:=\displaystyle\int_\Omega uv=0$ then $u=0$ ? where $H_0^1(\Omega)$ are the ...
0
votes
1answer
12 views

Characterization of product of dual dual Hilbert spaces.

Let $X_i$ be Hilbert spaces and $X_i'$ its dual spaces, with $i=1,2$. Let $F\in(X_1\times X_2)'$. Prove that exists $F_1\in X_1'$ and $F_2\in X_2'$ such that ...
2
votes
1answer
42 views

Spectral Measures: Scaled Spaces

Problem Given a Hillbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its probability measures by: ...
0
votes
1answer
83 views

Convergence of the spectrum under norm resolvent convergence

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in norm resolvent sense. Since $A_n \to A$ in strong ...
1
vote
1answer
50 views

Relation between residual spectrum and point spectrum.

Suppose T is a bounded operator on a Hilbert space. Show that if λ is in the residual spectrum of T, then $\bar{λ}$ is in the point spectrum of the adjoint. Here is what I think needs to be done. ...
3
votes
1answer
15 views

A sequence $L_n$ of compact bounded linear transformations on a hilbert space defines a convergent subsequence in each $L_n$ for a bounded sequence?

Let $L_n:\mathcal{H}\to\mathcal{H}$ be a sequence of compact bounded linear transformation on a Hilbert space $\mathcal{H}$, and $h_m$ be a sequence in $\mathcal{H}$. Since each $L_n$ is compact, ...
1
vote
1answer
33 views

Discrete Derivative: Closure?

Problem Given the Hilbert space $\ell^2(\mathbb{N})$. Consider the operators: $$T_0:\ell^2_0(\mathbb{N})\to\ell^2(\mathbb{N}):\quad T_0(a_k)_k:=(ka_k)_k$$ ...
2
votes
1answer
28 views

$\max <x,q>$ when $x \in H$ Hilbert and $q \in A^\bot$

How can I find $\max \langle x,q \rangle$ where $x$ is fixed in a Hilbert space $H$ and $q$ runs over $ A^\bot$ with $\|q\|=1$, $A$ a proper nontrivial subspace? (Here we assume $x \notin A, A^\bot$.) ...
0
votes
0answers
30 views

Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
0
votes
0answers
25 views

Step function integral inequality

I would like to prove the following inequality: $$\langle f,Id \rangle^2 \leq \langle f,1 \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i \mathbb{1}_{I(i)}(s)$, ...
1
vote
1answer
43 views

Spectral Measures: Analytic Elements

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the convergence radius by: ...
0
votes
2answers
28 views

Dense subspace of dense subspace is dense

Let $H$ be a Hilbert space and $V$, $W$ to linear subspaces such that $V\subset W\subset H$ with $V$ dense in $W$ and $W$ dense in $H$. Does this implie that $V$ is also dense in $H$? I think you can ...
1
vote
0answers
30 views

Is $H^1(\Omega, S^2)$ a Hilbert manifold?

I'm considering the topology of the function space $H^1(\Omega, \mathbb{S}^2)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain. Obivously it is not a vector space, but is it a Hilbert ...
0
votes
1answer
28 views

Spectral Measures: Nelson

Problem Given a Hilbert space $\mathcal{H}$. Consider a symmetric operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad T\subseteq\overline{T}\subseteq T^*$$ Denote the convergence radius by: ...
3
votes
2answers
33 views

Why is the additive category of Hilbert spaces not abelian

As an answer to this post Additive category that is not abelian it was said that the additive category of Hilbert spaces is not abelian. Why is that? Also what category of Hilbert spaces is this? ...
-1
votes
1answer
27 views

Reducing Subspaces: Characterization

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\mathcal{D}:=\mathcal{D}(T)$$ Regard a subspace: ...
0
votes
1answer
44 views

Reducing Subspaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad T=\overline{T}$$ Regard a closed subspace: ...
0
votes
1answer
28 views

Reducing Subspaces: Preliminary

Given a Hilbert space $\mathcal{H}$. Consider a dense domain: $$\mathcal{D}\leq\mathcal{H}:\quad\overline{\mathcal{D}}=\mathcal{H}$$ Regard a closed subspace: ...
1
vote
1answer
10 views

Linear functionals over a non dense subset in $\ell^2$

I have come across the following statement a couple of times, but cannot figure out quite how to justify it: If $A$ is not a dense subset of $\ell^2$ then its closure is not all of $\ell^2$ so ...
1
vote
1answer
13 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-Schwartz that ...
1
vote
1answer
40 views

Can an inner product on a vector space be negative?

This may be a noob question but I recently read a definition that an inner product on a complex vector space is said to be a positive-definite sesquilinear map. Doesn't positive definite mean that ...
3
votes
1answer
31 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
0
votes
0answers
22 views

Self adjoint operator with countable eigenvalues

It has been explained here that any self adjoint operator on a seperable Hilbert space inhabits at most countable many eigenvalues. Now I wonder, can one construct an operator $$ T : L^2([0,1]) \to ...
0
votes
1answer
30 views

Difference between hermitian and conjugate linear sesquilinear form

I am trying to work out the difference between hermitian and conjugate linear sesquilinear form. Let me elaborate on my confusion: Let $H$ be a Hilbert space. One definition (see e.g. here page 49) ...
0
votes
0answers
29 views

Is there a name for these sets of functions of several complex variables (most not analytic)?

For each $n \in \mathbb{N}$, I came up with the following sets that I found interesting; at least I've never seen them in the literature before. $S_n = $span{$z_1z_2 \cdots z_n, \bar{z}_1z_2\cdots ...
0
votes
1answer
22 views

The definition of continuity for linear functionals

I am trying to prove that a linear functional is continuous on the space $H^1(0,l)$, and I have a couple of different definitions. The one that I want to use is that $f$ is continuous if $f$ is ...
0
votes
0answers
40 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
0
votes
1answer
24 views

Isolated eigenvalues of a self adjoint operator

If $X$ is a separable Hilbert Space and $T : X \to X$ selfadjoint and bounded, then the point spectrum $$ \sigma_p(T) $$ is only countable as explained here. I have the following three questions: ...
1
vote
2answers
108 views

Every bounded sequence has a weakly convergent subsequence: salvage this proof?

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
1
vote
0answers
11 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
1
vote
1answer
21 views

Question about the following proof in Hilbert space

I started reading the book "Mixed Finite Element Technologies" by Peter Wriggers and Carsten Carstensen, and I have a question about the following. Here is the setup: Then, the authors prove the ...
1
vote
0answers
33 views

Is a cyclic subspace of a compact unitary representation finite dimensional?

Let $K$ be a compact Lie group and let $\rho_k: H \rightarrow H$ be a (strongly continuous) unitary representation of K on a Hilbert space H. Why does the orbit, $\rho(K)v$ ,of any $v\in H$ generate a ...
2
votes
1answer
30 views

Mourre Theory: Resolvent Formula

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its resolvent by: $$z\in\rho(H):\quad R(z):=(z-H)^{-1}$$ Introduce its ...
0
votes
2answers
35 views

Is the Norm of the Square Root of an Operator equal to the Square root of the Norm of the Operator

Suppose we have a positive operator $A \in \mathcal{B}(\mathcal{H})$, does it follow that $$\|A\|^{1/2} = \|A^{1/2}\|?$$ If not, is there some relation between these quantities?
1
vote
0answers
37 views

What are the negative-dimentional n-sphere and n-cube?

The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres. $$\begin{array}{ll} S_{n-1}(R) &= ...
1
vote
2answers
18 views

Why $\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$?

Let $f,g\in L^2$ with Lebesgue measure. and $K:L^2\to L^2$ be some linear and continuous operator. Show that $$\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$$ where $h\in H\subset L^2$.