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

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2
votes
3answers
604 views

Compact operators and uniform convergence

Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in ...
17
votes
2answers
4k views

Finding the adjoint of an operator

This is from my homework, I'm totally lost as to how to proceed. Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $(Tf)(x) = \int^x_0 f(s) \ ds$ What is the adjoint of $T$? ...
2
votes
1answer
248 views

Range of identity plus compact operator is closed

Suppose $K:H\to H$ is a compact linear operator on a Hilbert space $H$. How do I show that the range of $I+K$ is closed in $H$? I believe this is equivalent to showing that $\{x_n\}\subset H$ and ...
3
votes
1answer
188 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
1
vote
1answer
232 views

Direct sum of compact operators

I am having some trouble proving this: Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ ...
3
votes
2answers
572 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
1
vote
0answers
44 views

Radial and angular part of the space of compactly supported smooth functions

On page 124 of Thaller's The Dirac equation the following space is mentioned : $$C^{\infty}_0(0, \infty)\otimes C^\infty(\mathbb{S}^2)\subset L^2(0, \infty)\otimes L^2(\mathbb{S}^2),$$ where the ...
4
votes
1answer
124 views

Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...
5
votes
1answer
583 views

Showing the sum of orthogonal projections with orthogonal ranges is also an orthogonal projection

Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection. First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since ...
1
vote
0answers
267 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
0
votes
1answer
107 views

Tensor product of Hilbert Algebras

A Hilbert algebra is an inner product space that is also a *-algebra where the various operations and structures interact according to some axioms. One of those axioms is that the linear operation ...
0
votes
1answer
152 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
4
votes
3answers
737 views

Question about example of non-separable Hilbert space

I have come across the following example of a non-separable Hilbert space: Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ ...
1
vote
1answer
141 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
4
votes
5answers
887 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
4
votes
2answers
152 views

Do all angles occur in Hilbert spaces?

Let $X$ be a Hilbert space with scalar product $(\cdot,\cdot)$. Then for two vectors $v,w$ of norm $1$, we can interpret $(v,w)$ as an angle, so that $(v,w)=\cos(\varphi)$ for a unique angle ...
2
votes
1answer
169 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} ...
4
votes
1answer
59 views

How to conclude $\Re $ is zero?

I'm in a Hilbert space $H$ and for $z,v, h \in H$ and $t \in \mathbb C$ I have $$ \|z\|^2 \leq \|h−(tv+y)\|^2 = \|z−tv\|^2 =\|z\|^2 −2\Re(t⟨v,z⟩)+|t|^2\|v\|^2$$ According to my notes it follows ...
1
vote
2answers
383 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
1
vote
2answers
153 views

Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)

Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that ...
2
votes
2answers
121 views

proving “$C^1([−1,1])$ is dense in the given space with given norm”

Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I ...
8
votes
5answers
990 views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
3
votes
1answer
177 views

A problem on Fourier transforms and orthogonality

Let $f$ be a square integrable function, strictly positive almost everywhere. Consider the family of functions $f_a=f(x+a)$, where $a$ is any real number. I want to prove that if a function is ...
2
votes
2answers
100 views

Minimizing a functional on $L^2$

Let $$ \mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx = \int_0^\pi f(x)\sin x dx = 1\right\}. $$ Solve this problem: $$ \tag{P} \min_{\mathcal M} \int_0^\pi ...
6
votes
0answers
388 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
1
vote
1answer
157 views

norm of operator in Hilbert space and complex conjugate Banach space

Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex ...
3
votes
1answer
132 views

How quickly does the inner product of an L-2 function against its translates decay?

Let $H$ be the Hilbert space $L^2(\mathbb{R})$. For $t \in \mathbb{R}$, let $\lambda_t \in B(H)$ be the unitary operator which translates by $t$, that is $(\lambda_t \xi)(s) = \xi(-t +s)$. For $\xi ...
2
votes
1answer
75 views

Application of Banach Seperation theorem

Let $(\mathcal{H},\langle\cdot,\cdot\rangle)$ be a Hilbert Space, $U\subset \mathcal{H},U\not=\mathcal{H}$ be a closed subspace and $x\in\mathcal{H}\setminus U$. Prove that there exists ...
-1
votes
1answer
146 views

Stone-Cech compactification of the separable Hilbert space

Where can I read about the Stone-Cech compactification of the separable Hilbert space?
3
votes
1answer
128 views

Dense subspace of $\ell^2$

Is the set \begin{align} A=\left\{a=(a_1,a_2,\dots)\in\ell^2 \ \ \lvert \ \ \sum_{k=1}^\infty \frac{a_n}{n}=0 \right\}\subset\ell^2 \end{align} dense in $\ell^2$ Is the following argument ...
3
votes
1answer
201 views

Compact operators between Hilbert spaces

I have the suspect that the following statement is true, but I don't how to prove it. Any suggestion? Thanks to all! Let $X$, $Y$ be Hilbert spaces and let $T \colon X \to Y$ be a linear continuous ...
4
votes
1answer
314 views

A linearly independent, countable dense subset of $l^2(\mathbb{N})$ [duplicate]

Possible Duplicate: Does there exist a linear independent and dense subset? I am looking for an example of a countable dense subset of the Hilbert space $l^2(\mathbb{N})$ consisting of ...
3
votes
2answers
326 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
408 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
109 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
205 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
281 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
192 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
233 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
345 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
495 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
107 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
138 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
544 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
158 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
349 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 ...