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

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15 views

Closed subspace of weighted L2 space?

Let $L_{w_{\xi}}^{2}[0,\infty)$ be a weighted $L^2$-space with weight function $w_{\xi}(x) = \frac{\exp\left({-(x+\xi)^3}\right)}{(x + \xi)^2},\; \xi > 0$ and let $T$ denote the operator that ...
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1answer
29 views

A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
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1answer
41 views

Mourre Adjoint: Bounded Maps (II)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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39 views

$||f||_1 =(\int_a^b [|f|^2+|f'|^2]dx)^{1/2}$. Is this normed space complete?

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$||f||_1 =\left(\int_a^b \left(|f|^2+|f'|^2\right) dx \right)^{1/2}$$ Is this normed space ...
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1answer
49 views

Mourre Adjoint: Bounded Maps (I)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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1answer
30 views

Proof of positive definiteness

$Lu = -u'' + c u$ where c is some constant The question is when it's positive definite in square integrable on $[0; 1]$ with $u(0)=u(1)=0$ $(Lu, u) = \int^1_0 u Lu dx = -u u''+c u^2 dx = \int^1_0 ...
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9 views

Matrix of $T$ is triangular in Hilbert space

Could any one tell me how to solve this problem? $T $be a linear transformation on $H$ , we need to show there exists a basis $B$ relative to which the matrix of $T $ is triangular, if $T$ is normal ...
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1answer
6 views

Chain of closed linear subspace

Could anyone tell me how to solve this problem? $T$ be any operator on Hilbert space $H$, we need to show that there exists closed linear subspaces $M_1,\dots, M_n$ such that $\{0\}\subseteq ...
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1answer
80 views

Is $\operatorname{span}\varepsilon=\overline{\operatorname{span}\varepsilon}$ in Hilbert Space?

The term "span" is defined in linear span. So $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbf{K}} \right \}$, and ...
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1answer
23 views

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$.

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$. Hint: Take a Cauchy sequences $(x_r)$, where $x_r=\sum ...
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2answers
39 views

Strong convergence from weak convergence

I am trying to show that a sequence $(x_n)_n \subseteq \mathcal{H}$ converges strongly to $x$ if it converges weakly to $x \in \mathcal{H}$ and $\|x_n\| \to \|x\|$ as $n \to \infty$ $\mathcal{H}$ is ...
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2answers
54 views

Is the unit sphere in an infinite dimensional Hilbert space closed?

Is a unit sphere in an infinite dimensional hilbert space closed. By the triangle inequality it is clear that the all the limit points of the sphere are inside the closed unit ball. But I cannot ...
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1answer
75 views

Why do dagger categories supposedly capture the structure of a Hilbert space?

A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ...
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1answer
32 views

Spectral Measures: Constructions

Any constructions are welcome!!! Given a Hilbert space $\mathcal{H}$. Denote projections by: $$\mathcal{P}(\mathcal{H}):=\{P\in\mathcal{B}(\mathcal{H}):P^2=P=P^*\}$$ Consider spectral measures: ...
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1answer
30 views

Compatibility of topologies and metrics on the Hilbert cube

Consider the Hilbert cube $Y = [0,1]^\mathbb{N}$. It is easy to define four classes of metrics on $Y$ for $\gamma>0$ and $\omega>1$: $$d^\gamma_{sup,pol}(x,y) = \sup_{k\geq 1} ...
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1answer
52 views

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$.

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$. I know that the orthogonal complement of $X$ is the set ...
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1answer
22 views

Spectrum of double infinite shift using isometry to Fourier series

I'm trying to find the spectrum of the operator $T: l^2(\mathbb{Z}) \to l^2(\mathbb{Z})$ given by right shift but I am having some difficulties. I can show that $l^2$ is isomorphic to ...
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2answers
24 views

To show $I+ A$ is non singular

$A$ is a positive operator on Hilbert space $H$, I have to show the title of this question. Since $A $ is positive so all eigenvalues are $\ge 0$, so eigenvalues of $I+A$ are $\ge 1$, so $\det(I+A) ...
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1answer
54 views

Riesz-Fischer theorem

The aim of this exercise is to prove the Riesz-Fischer theorem for Hilbert spaces that aren't separable. Let $I$ an index set and $1\leq p \leq \infty$. Let $\mathcal{F}=\{F\subset I: F$ is ...
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1answer
23 views

Given any countable collection of non-zero vectors in a Hilbert space

Let $\{\alpha_i\}$ be a countable collection of non-zero vectors in a Hilbert space $H$. Is there exist a vector $\beta \in H$ such that $\langle \beta , \alpha_i \rangle \neq 0$ for all $i$ ?
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13 views

Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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2answers
42 views

Every non-compact Hermitian operator P has an infinite dimensional invariant subspace on which P is bounded from below

I want an explanation of the following statement. If $P$ is a Hermitian operator on Hilbert space and not compact, there exists an infinite-dimensional subspace $M$, invariant under $P$, on which $P$ ...
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2answers
50 views

Sufficient condition for $0$ to be in a closed convex hull in a Hilbert space.

I am working on the following problem: Let $\mathcal{H}$ be a Hilbert space, let $\left\{a_n\right\}_{n=1}^\infty \subset \mathcal{H}$ be a sequence such that $||a_n|| = 1$, and consider the ...
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2answers
33 views

uniformly convergent subsequence of bounded linear operators on a Hilbert space?

I am working a problem in which we start with a Hilbert space $\mathcal{H}$ and a sequence $\left\{a_n\right\} \subset \mathcal{H}$ with $||a_n|| = 1$. We also assume that $$\lim_{n \to \infty} ...
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1answer
22 views

The real version of the Cuntz algebra

Assume that $H$ is a real separable Hilbert space. Are there two operators $T,S \in B(H)$ which satisfy $$TT^{*}+SS^{*}=1,\;\;T^{*}T=S^{*}S=1$$ where * is the adjoint operator?
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18 views

Eigenvalue-eigenvector equation for an operator

Proof: Given an eigenvalue-eigenvector equation, suppose that the state vector depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t. ...
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0answers
10 views

Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
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1answer
19 views

Wave Operators: Reducibility

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_0:\mathcal{D}(H_0)\to\mathcal{H}_0:\quad H_0=H_0^*$$ $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and a bounded ...
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31 views

If $\forall v \in V, \ a(Tu,v)=(u,v)$ is $T$ a bounded an regular operator?

Let $V, H$ two Hilbert spaces infinite dimensional. If the bilinear form $a(.,.)$ satisfies There exists a constant $\alpha>0$ such that $\forall v \in V, \ a(v,v)\geq \alpha \|v\|^2.$ There ...
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0answers
20 views

Relation between discrete and continuous inner product of a function

Let $g_1^d$ be descrete (sampled) version of continuous function $g_1$ , same for $g_2$. So we have $$\left<g_1^d,g_2^d\right>=\sum_{n=-\infty}^\infty {g_1^d[n] ...
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24 views

Finding isometries of a Banach Spaces.

Given a Hilbert Space $(H,\langle,\rangle)$, $x,y\in H$ and $D\subset H$ a subspace of $H$ (I mean, the operators $+$, $\cdot$ and $\langle,\rangle$ in D are the restrictions of the respective ones in ...
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0answers
19 views

Schauder Basis and Fourier series

I'm looking at the constuction of the Brownian Motion given by Lévy-Ciesielki. We want to use Haar functions as basis of $L^2([0,1],\mathcal{B},\lambda)$. So on the n-th partition of $(0,1]$ ...
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1answer
33 views

Prob. 15, Sec. 3.10 in Kreyszig's functional analysis book: $\Vert T^2 \Vert =\Vert T \Vert^2$ if $T$ is normal?

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a bounded linear operator, and let $T^*$ denote the Hilbert adjoint operator of $T$. I can show that if $T$ is normal (i.e. $T T^* = T^* T$), ...
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1answer
28 views

Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
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37 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
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1answer
41 views

Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace.

Show that if a vector subspace of a Hilbert space is closed, then it is a Hilbert subspace. Here is what I have so far. Any comments or hints are greatly appreciated. Let $H$ be a Hilbert space and ...
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1answer
39 views

Find an inner product that makes a given set of linearly independent vectors orthogonal

I need to find an inner product such that given a set $S$ of linearly independent vectors in a Hilbert space $H$, $S$ will be orthogonal with these product. I thought Gram -Schmidt Process would help ...
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0answers
28 views

$Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$ L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}. $$ This operator is not ...
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34 views

Exotic applications of Hilbert spaces?

So my final exam for an introductory course on Hilbert spaces is just a weeks away. I enjoyed the course, we covered the theory in enough detail to illustrate its richness and elegance. I'm aware of ...
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1answer
70 views

Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$ [duplicate]

I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$. We know trivially that $0 \in L^2(0,1)$. I need to show that $\langle u^*,\sin(kx) \rangle \to \langle ...
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2answers
60 views

Show that there exists no positive continuous function $f$ defined on $[a,b]$ that satisfies the following conditions:

Show that there exists no positive continuous function $f$ defined on $[a,b]$ that satisfies the following conditions: $\int_a^bf(t)dt=1$, $\int_a^btf(t)dt=\alpha$, $\int_a^bt^2f(t)dt=\alpha^2$ ...
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1answer
33 views

Subsec. 4.1-8 in Kreyszig's functional analysis book: Does every inner product have a total orthonormal set?

In every Hilbert space $H \neq \{0 \}$, there exists a total orthonormal set. I think I've understood the proof given by Erwin Kreyszig in Introductory Functional Analysis With Applications. ...
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0answers
48 views

An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
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1answer
55 views

Prob. 10, Sec. 3.9 in Kreyszig's functional analysis book: The null space and adjoint of the right-shift operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$, let $T \colon H \to H$ be defined as follows: Since span of $(e_n)$ is dense in $H$, for every $x \in H$, we have $$x = ...
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24 views

Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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1answer
57 views

Prob. 4, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Image of a set under the adjoint operator

Here's Prob. 4, Sec. 3.9 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $H_1$ and $H_2$ be Hilbert spaces, and let $T \colon H_1 \to H_2$ be a bounded linear ...
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0answers
23 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
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26 views

Prob. 2, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Inversion and adjointness

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bijective bounded linear operator whose inverse is bounded. Then how to show that $(T^*)^{-1}$ exists and $$(T^*)^{-1} = (T^{-1})^*?$$ My ...
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0answers
57 views

Classical solution involving semigroups

Let $\{T_{D}(t)\}_{t\ge 0}$ a $C_{0}$-semigroup with generator $A+D$, with $D\in\mathcal{L}(X)$ and let $x_{0}\in \mathcal{D}(A)$. I want to show that ...
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42 views

Prob. 8, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hilbert space is isomorphic to its second dual

How to show that any Hilbert space $H$ is isomorphic to its second dual space $H^{\prime\prime} = (H^\prime)^\prime$? (This is Prob.8, Sec. 3.8 in Erwine Kreyszig's Introductory Functional Analysis ...