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

learn more… | top users | synonyms

1
vote
1answer
16 views

Proof of Hilbert Projection Theorem

If M is a closed subspace of the Hilbert space H and $x \in H$, then: There exists a unique element $\hat{x} \in M$ such that: $\|x-\hat{x} \|=\inf_{y \in M}\|x-y \|$ To proof of the existence of ...
0
votes
0answers
15 views

Does conjugation by half invertible matrices preserve spectrum?

Conjugation by an invertible matrix preserves the spectrum, but does conjugation by a left/right invertible matrix also preserve spectrum? My motivating situation was considering non-unitary ...
1
vote
1answer
27 views

Spectral Measures: Adjoint

This thread is only Q&A! (See the hint: SE: Q&A) Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the ...
0
votes
1answer
11 views

Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
1
vote
1answer
29 views

Riesz Representation Theorem in Wikipedia vs. Rudin's RCA

In Rudin's Real & Complex Analysis theorem 2.14, the Riesz representation theorem gives (in my very rough phrasing) an injection from linear functionals on a space to positive Borel measures which ...
1
vote
0answers
18 views

Total set in a Hilbert space

Definition: A subset of a Hilbert space is total if its span is the entire space. Halmos in his book (A Hilbert space problem book) asks below question: There exists a total set in a Hilbert ...
2
votes
0answers
23 views

Dimension of a Hilbert space

Halmos in his book (A Hilbert space problem book) says, 1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. 2- Also he proves if orthogonal dimension of ...
1
vote
1answer
25 views

Spectral Measures: Boundedness

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
0
votes
1answer
57 views

Spectral Measures: Existence

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad ...
0
votes
1answer
34 views

Spectral Measures: Invertibility

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
0
votes
1answer
25 views

Why can we consider elements of a normed space $X$ as elements of a normed space $Y$, if there is an embedding between these spaces?

Let $(X,\left|\;\cdot\;\right|)$ and $(Y,\left\|\;\cdot\;\right\|)$ be normed spaces and $\iota :X\hookrightarrow Y$ be an embedding. Often when I read that such an embedding $\iota$ exists, I read ...
1
vote
1answer
47 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
0
votes
1answer
49 views

Mourre Adjoint: Bounded Maps (III)

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: ...
0
votes
1answer
31 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
0
votes
1answer
26 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
4
votes
4answers
61 views

Is every projection on a Hilbert space orthogonal?

I'm highly doubtful that the answer is "yes," but I fail to see what's incorrect about this very basic proof I've thought of. If someone could point out my error, I'd appreciate it. My logic is as ...
0
votes
2answers
35 views

Prove uniform convergence of $\sum_{e \in \xi} \langle Th, e \rangle e$ for $\| h \| \leq 1$

Let $\xi$ be a basis for Hilbert space $H$. From Parseval's Identity, for every $x \in H$ we have $x = \sum_{e \in \xi} \langle x, e \rangle e$. Thus, for every bounded operator $T : H \rightarrow H$ ...
1
vote
1answer
30 views

If $AT = TA$ for every continuous compact operator $T$, then $A$ is a multiple of identity

Given a Hilbert space $H$, let $A: H \rightarrow H$ be a bounded operator. Show that if $AT = TA$ for every continuous compact operator $T : H \rightarrow H$, then $A$ is a multiple of identity ...
1
vote
0answers
50 views

Show that every continuous finite rank operator $T$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$

Can someone help me with this question? Suppose that $H$ and $K$ are Hilbert spaces. Show that every operator $T \in B_{00}(H, K)$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$, where ...
1
vote
0answers
40 views

Prove that $\sin(n\pi x)$ weakly converges to $0$ in $L^2(0,1)$ [duplicate]

Let $$f_n(x):=\sin(n\pi x)\;\;\;\text{for }x\in (0,1)$$ and $$\langle f,g\rangle:=\int_{(0,1)}fg\;d\lambda^1\;\;\;\text{for }f,g\in L^2(0,1)$$ I want to show, that $(f_n)_{n\in\mathbb{N}}$ weakly ...
4
votes
1answer
66 views

Show that $\{ x_n \} \overset{T}{\mapsto} \{ \sum_{k=1}^{\infty} a_{nk} x_k \}$ is compact

Can someone help me with this question? Let $\ell^2$ be the space of complex sequences $\{ x_1, x_2, \ldots \}$ that $\sum_{n=1}^{\infty} \lvert x_n \rvert ^2 < \infty$. If $\mu$ be Counting ...
1
vote
0answers
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 ...
1
vote
1answer
30 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 ...
1
vote
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: ...
2
votes
1answer
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 ...
0
votes
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: ...
1
vote
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 ...
0
votes
0answers
11 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 ...
0
votes
1answer
8 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
2answers
40 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 ...
0
votes
2answers
64 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 ...
5
votes
1answer
83 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 ...
0
votes
1answer
46 views

Spectral Measures: Constructions

Any constructions welcome!!! Given a Hilbert space $\mathcal{H}$. Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ That are additive: ...
2
votes
1answer
35 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} ...
1
vote
1answer
57 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 ...
0
votes
2answers
33 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 ...
1
vote
2answers
27 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) ...
2
votes
1answer
56 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 ...
0
votes
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 $i$ ?
1
vote
0answers
15 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, ...
1
vote
2answers
46 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 $P$ ...
3
votes
2answers
52 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 ...
2
votes
2answers
36 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} ...
0
votes
1answer
24 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?
1
vote
0answers
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. ...
0
votes
0answers
12 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 ...
0
votes
1answer
43 views

Wave Operators: Reducibility

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
0
votes
0answers
32 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 ...