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

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3
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2answers
317 views

norm of a normal operator using projections

Let $H$ be a Hilbert space and $T$ a normal operator on $H$. In the sequel, ${\rm tr}$ denotes the trace for trace class operators. Do we have $$ \vert\vert T \vert\vert= \sup |{\rm tr} (TP)| $$ ...
11
votes
1answer
400 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
0
votes
1answer
63 views

a representation condition of Hilbert space

Let $H$ be a Hilbert space over $\mathbb{C}$. Let $\phi_{j} \in H$ and let $c_j$ be scalars. Prove that there exists a representation $$f=\sum_{j}c_{j}\langle f,\phi_{j}\rangle\phi_{j}, \forall f\in ...
1
vote
0answers
108 views

Hilbert Spaces and Projections

Suppose that $\{Y_{t}: t \in \mathbb{Z} \}$ is a stationary zero mean time series. Consider the Hilbert space $\mathcal{H}$ generated by the random variables $\{Y_t: t \in \mathbb{Z} \}$ with inner ...
1
vote
1answer
203 views

A question on self adjoint operator and numerical range.

Let $H$ be a complex Hilbert space and $T$ be self-adjoint operator. Prove that $$ \Vert T\Vert = \sup\{|\lambda|:\lambda \in W(T)\} $$ We are supposed to use the following exercise and the fact that ...
0
votes
1answer
269 views

Closed linear span of a frame in a Hilbert space $\mathcal{H}$ coincide with $\mathcal{H}$

Definition of the problem Let $\mathcal{H}$ be a separable Hilbert space and $J\subset\mathbb{N}$ an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$ be a frame for ...
2
votes
2answers
188 views

Perturbation theorem of Weyl

Does anyone know where to find something about the perturbation theorem of Weyl, preferably on the internet. The theorem I'm talking about states: let $A$ be a self-adjoint operator on a Hilbert ...
1
vote
1answer
228 views

Analysis operator $T_\Phi$ is injective and has a closed range

Definition of the problem Let $\mathcal{H}$ be a separable Hilbert space on $J\subset\mathbb{N}$ an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$ be a frame for ...
1
vote
1answer
100 views

References on Algebraic Operators

Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$. In ...
1
vote
1answer
338 views

Numerical range of an operator on Hilbert spaces.

If $H$ is a Hilbert space and $T$ is in $\mathcal{L}(H)$, the numerical range of $T$ is defined by $$W(T) := \left\{(Tx; x) \mid x \in H,\ \|x\| = 1 \right\}.$$ We have to prove that The point and ...
2
votes
2answers
482 views

Direct sum of Hilbert Space subspaces - Notation?

This is a really basic question sorry, I just need to make sure I have my understanding correct. Given an infinite dimensional Hilbert space $\mathcal{H}$ and two subspaces, also Hilbert spaces, ...
3
votes
1answer
106 views

Example of an Hilbert space operator

There is a theorem in functional analysis, that says that for a selfadjoint compact operator $T:H\rightarrow H$, either $\lVert T\rVert $ or $-\lVert T\rVert$ is an eigenvalue. For finite dimensional ...
1
vote
1answer
57 views

An explicit example of an invariant halfspace of the unilateral shift?

In a recent talk, A. Popov stated the following fact The unilateral shift on $\ell^2$ has invariant halfspaces. Halfspaces are closed subspaces whose dimension and codimension are both infinite. ...
1
vote
1answer
137 views

Reproducing Kernel of subspace of $L^2(0,1)$

Definition of the problem Let $\mathcal{H}$ be a Hilbert space which consists of functions, defined on a set $S$. Let $k:S\times S \rightarrow \mathbb{K}$ is our reproducing kernel for $\mathcal{H}$. ...
2
votes
1answer
528 views

Sesquilinear forms on Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $B:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}$ be a sesquilinear form. Prove that TFAE: $(i)$ $B$ is continuous. ...
3
votes
1answer
59 views

Convergent operatorial series

An exercise I was doing asks (among other things) for the values of $z\in\mathbb{C}$ for which the following (operatorial) series converges absolutely: $$\sum_{n=0}^{\infty}z^nA^n$$ where $A$ is an ...
2
votes
2answers
156 views

I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
10
votes
1answer
347 views

Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections. The ...
3
votes
2answers
97 views

Equiangular sequence in $\ell^2$

I have a countable infinite normed, equiangular sequence $a_n \in \ell^2$, i.e $\langle a_n, a_m \rangle=\theta$ for $n\not=m$ and $\langle a_m, a_m \rangle =1$ for some $\theta <1$. It's clear ...
1
vote
2answers
66 views

For any sequence in $L^2$ there is a function in $L^2$ s.t. is not orthogonal to any point of the sequence

How to prove that for any sequence $(f_n) \subset L^2[0,1]\setminus \{0\}$ there is a function $g \in L^2[0,1]$ such that $$\int f_n g dx \neq0\ \forall n\geq 1?$$ I tried to use a weak limit of ...
4
votes
4answers
992 views

Weak and pointwise convergence in a $L^2$ space

Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$. Assume that the sequence $(f_n)$ converge pointwise ...
1
vote
1answer
90 views

Operator norm is not induced by a scalar product

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, $\dim\mathcal{H}\geq2$. Prove that the operator norm on $L\left(\mathcal{H}\right)$ is not induced by a scalar product. We are hinted ...
2
votes
2answers
514 views

Weak convergence in Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $\left(x_{n}\right)_{n\in\mathbb{N}}\subset\mathcal{H}$ be a sequence. Prove the following: If ...
5
votes
0answers
377 views

Sum of operator and adjoint is self-adjoint

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
2
votes
0answers
271 views

Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?

It seems not to be a easy problem. I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
1
vote
2answers
177 views

Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?

I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...
3
votes
1answer
371 views

Estimate for Operator Norm in Hilbert Space

Let $H$ a hilbert space with an orthonormal basis $(e_n)_{n\in \mathbb{N}}$ and $F$ a linear operator, such that $\langle e_k,F e_n\rangle =:\phi(n,k)$. Find a good estimate for $\lVert F\lVert$ in ...
2
votes
1answer
157 views

Compute operator norm by image on orthonormal basis

Let $e_n$ a orthonormal basis for a Hilbert space and $T$ a bounded linear operator. Is the following correct? $$\lVert T \lVert^2 \leq \sup_{n \in \mathbb{N}} \sum_{k \in \mathbb{N}} |\langle ...
0
votes
2answers
110 views

Net of Projections

I am trouble proving the following proposition in Conway's functional analysis book. $H$ is an arbitrary Hilbert space, $I$ is an index set. Prop - Let $\{P_i:i\in I\}$ be a family of pairwise ...
3
votes
2answers
435 views

A continuity condition for a bilinear form on a Hilbert space

Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be bilinear and symmetric. Suppose there is a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be ...
4
votes
6answers
580 views

Bounded operator that does not attain its norm

What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred. All of the simple examples I have looked at (the identity operator, the ...
2
votes
2answers
124 views

Hilbert space linear operator question

Let $\mathcal{H}$ be the vector space of all complex-valued, absolutely continuous functions on $[0,1]$ such that $f(0)=0$ and $f^{'}\in L^2[0,1]$. Define an inner product on $\mathcal{H}$ by ...
3
votes
1answer
112 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
2
votes
1answer
109 views

Family of Self-Adjoint Operators that are Multiplications on a Common $L^2(\mu)$?

Suppose that $H$ is some (complex) Hilbert space and that $\{T_\alpha: \alpha \in I\}$ is some collection of bounded self-adjoint operators on $H$. A version of the spectral theorem states that for ...
2
votes
1answer
460 views

Bessel sequence in Hilbert space

I'm posting this question again because I'm still confused about the answer! A sequence $\{f_{n}\}_{n\in I}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that ...
0
votes
1answer
92 views

Convergence of a sequence of sets

Given a squence of sets $\{S_{n}\}_{n=1}^{\infty}$, where each $S_{n}$ is countable set. I came over this statement in some article, it says: "Let $S$ be the weak limit of $S_{n}$". But I couldn't ...
2
votes
1answer
217 views

Projections on Hilbert space

My question is: Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a projection if and only if $T$ is the identity on the orthogonal complement of its kernel. Thanks
0
votes
1answer
107 views

If a sequence is not a frame

A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...
1
vote
2answers
854 views

Inner product on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$, respectively. Then $H_1\otimes H_2$ is at least a pre-Hilbert space (we are ...
2
votes
1answer
84 views

Hilbert space question

Let $\{x_n\}$ be a sequence of pairwise orthogonal vectors in a Hilbert space $H$. Prove that the following are equivalent: a) $\displaystyle\sum_{n=1}^\infty \|x_n\|^2<\infty$ b) ...
3
votes
1answer
483 views

Bounded linear operator on a Hilbert space

I am having a bit of difficulty with the following homework problem. Let $\{x_n\}$ be an orthonormal basis in a Hilbert space $V$ over $\mathbb{C}$ and let $\{c_n\}_{n \in \mathbb{N}}$ be a fixed ...
3
votes
1answer
85 views

Function space in QM

I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
0
votes
1answer
466 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
2
votes
1answer
105 views

A question about the proof in functional analysis

I'm now reading Pazy's book about the semi-group operator. To prove the existence of the solution of KdV equation. He define the Hilbert space $H^s(\mathbb{R})$ $$ \Vert ...
1
vote
3answers
1k views

Question on weak convergence ( Example).

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
0
votes
1answer
251 views

Orthonormal basis implies orthogonal basis!

If $\{\frac{f_{n}}{\|f_{n}\|}\}_{n\in I}$ is an orthonormal basis for a separable Hilbert space $H$, and $\{f_{n}\}_{n\in I}$ is a complete and orthogonal set in $H$, is it true that $\{f_{n}\}_{n\in ...
5
votes
2answers
965 views

How to show a compact, closed-range operator on an infinite-dimensional Hilbert space has finite rank, without using the open-mapping theorem?

If $H$ is an $\infty$-dimensional Hilbert space and $T:H\to{H}$ is a compact operator with closed range, how do I show that $T$ has finite rank, without using the open-mapping theorem? (The ...
2
votes
1answer
235 views

Why are only Sobolev spaces with certain exponents Hilbert Space?

I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ . Here $||u||_{W^{k,2} (\Omega)} $ = ...
1
vote
1answer
545 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
3
votes
2answers
333 views

The image of orthonormal basis under compact operator

I need a help to prove that statement: if $\{e_n\}$ an orthonormal basis in Hilbert space $H$ and $A$ is a compact operator from $H$ to $H$, then $Ae_n\rightarrow 0$. Thx for any help.