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

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32 views

Can A Weakly Convergent Sequence Be Unbounded in Norm

Let $H$ be a complex Hilbert spaces and let $(x_n)$ be a sequence in $H$ which converges to $0$ weakly. Question. Can $(x_n)$ be unbounded in the norm? In other words, is it possible that the ...
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28 views

Spectrum of nonnegative operator

Let $A$ be a bounded, nonnegative operator on a complex Hilbert space $H$. Prove that the spectrum $$\sigma(A)\subset[0,+\infty].$$ We say that an operator $A$ is nonnegative if it is self adjoint and ...
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24 views

Are Isomorphic Hilbert Spaces Still Solutions

So I was wondering say that you have solutions to some ODE, $y_n$ with eigenvalues of $\lambda_n $. Well, the linear combinations of these solutions are also solutions to the original ODE ( i.e. $ ...
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1answer
39 views

Define $T_n \in B(H)$ as $T_n(x)= \langle x,e_1\rangle e_n$ Show that $T_n$ converges in weak operator topology but not strongly.

In some notes I found the following claim.Can someone please help me in proving this? Let $H$ be a separable Hilbert space and $(e_n)$ be ONB for $H$. Define $T_n \in B(H)$ as $T_n(x)= \langle ...
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2answers
30 views

Suppose $(x_n)$ is bounded and for all $e \in E$, $<x_n,e> \to 0$ as $n \to \infty$.Does this implies that $<x_n,y> \to 0$ for all $y$ in $H$?

I'm reading a solution in which following result is used. Suppose $H$ be a hilbert space and $E$ be a ONB for H.Suppose $(x_n)$ is bounded and $<x_n,e> \to 0$ as $n \to \infty$ for every $e ...
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1answer
19 views

Uncountable linearly independent set of vectors implies the space is not separable? + A question on self adjoint operators

Let $\mathcal H$ be a hilbert space and let $A\subset \mathcal H$ be an uncountable set of linearly independent vectors. Does this imply $\mathcal H$ is not separable? If $A$ was a set of orthonormal ...
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1answer
18 views

Let $(e_n)$ be ONB for $H$. Define $T_n \in B(H)$ as $T_n(x)= \langle x,e_n\rangle e_1$. Show that $T_n$ converges strongly but in Norm.

I was looking for a example for sequence of bounded operators which converges strongly but does not converge in Norm. I've found the following example somewhere on internet, but unable to prove this. ...
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1answer
67 views

Dimension of a subset of $\ell_2$

I am wondering whether the following space is a finite-dimensional subset of $\ell_2$ or an infinite-dimensional one? $$S(\ell_2)= \{ (x_1,x_2,\ldots) \in \ell_2 \mid \sum_{n=1}^\infty (nx_n)^2 ...
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1answer
52 views

Orthogonal projection

Given the Hilbert space $H=\mathbb{L}^2 ([0,1])$ endowed with the canonical scalar product $\langle f,g \rangle = \int_0^1 f(x)g(x)dx$, as well as the orthogonal projection $p_n:H\mapsto H$ defined ...
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37 views

Is intersection of a dense subspace and a closed subspace of a Hilbert space also Dense?

I have a Hilbert space $H$ and a closed operator $T$ defined on its domain $D(T)$ which is dense in H. Also $M = \text{range} \ T^n$, for some $n$, is given to be closed. Consider the restriction of ...
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12 views

Implicit feature space of Power Kernel

For the polynomial kernel, $K(x,y) = (x^Ty+c)^d$, the implicit feature space $\phi$ for which $K(x,y) = \phi(x)^T \phi(y)$ is of finite dimension and well known [1][2]. It is also well known that the ...
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1answer
10 views

Relatioship between Weight of inner product on Hilbert space and elements

So I was wondering say you have a Hilbert such as the inner product is defined as $$ <f,g>= \int fge^{-x^2}dx $$ where $ f,g \in M $ where $M$ is the set of functions on which the inner product ...
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28 views

Questions on the Riesz-Fréchet theorem.

There are two points in the Riesz-Fréchet theorem that I am unsure about. Consider the following statement and proof for the Riesz-Fréchet (Riesz Representation) theorem: Theorem: "Let H be a ...
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15 views

How do we take the limit of this quantum operation?

I am wondering how to take the following limit: \begin{align} L= \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} dy \, \left(1 - \frac{1}{\sqrt{ \pi} \sigma } ...
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1answer
35 views

Eigenvalues of the Hilbert - Schmidt operator

Ok so the question is very simple, if you have the Hilbert Schmidt operator: $$Kf[x]=\int_a^b k(x,y)f(y)dy,$$ with $f\in L^2(a,b)$, how can you find his eigenvalues(i.e, $Kf_n=\lambda_n f_n$)? You ...
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1answer
31 views

Finding hilbert space basis

I already have two vectors expressed in basis $\{|u_1 \rangle, |u_2\rangle, |u_3\rangle\}$: $$|\phi_1\rangle = \frac{1}{\sqrt{3}}(|u_1\rangle + i |u_2\rangle - |u_3\rangle) $$ $$|\phi_2\rangle = ...
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1answer
55 views

In an orthogonal sequence is weakly convergent, then it is convergent

Let $ H $ be a Hilbert space, and let $\{x_k \}_{k\in \mathbb{N}}$ be an orthogonal subset of $H$. If for every $y\in H$, $\sum \left<x_k, y\right>$ converges, then $ \sum x_k $ converges ...
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18 views

Lumer-Phillips Theorem for non-contraction semigroups?

Let $H$ be a closed operator on a Hilbert space $\mathcal H$. The Lumer-Phillips theorem states that $H$ is a generator of a one-parameter contraction semigroup if and only if $\Re\langle ...
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20 views

Prove that $\bar{A}^\perp = A^\perp$

This is a rather easy exercise, but I'd like to be sure that I properly proved it. Given Hilbert space $X$ and a subset $A \subset X$, we have:$$\bar{A}^\perp = A^\perp$$ Proof: Since $A ...
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21 views

Is such a function in $L^2$?

Let $L^2$ denote $L^2([0,T];\mathbb{R}^N)$ and let $A:[0,T]\times L^2 \to (L^2)^\prime =L^2$ be a Carathéodory function satisfying the following estimate: $$\exists \alpha \in ...
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30 views

Existence of Unitary Map

I've been recently introduced to Unitary operators of a Hilbert space and I've been wondering the following. Existence of a unitary operator $T$ on a (possibly infinite) Hilbert space $H$ is simple ...
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37 views

The distance is attained by a unique point

Theorem: Let $K$ be a convex and closed subset of a Hilbert space $X$ and $x \in X$. Then there is a unique $y_x \in K$ such that $$\|x-y_x\|=d(x,K):=\inf \{\|x-y\|: y \in K \}$$ Remarks: if $K$ ...
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1answer
26 views

Eigenfunction representation of the L2 derivative

I think the main idea of the definitions that follow is to define some sort of generalized double derivative on a subset of $L^2[0,1]$ Define $D(K)$ to be the subset of $C^1[0,1]$ made up of ...
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166 views

Is every Banach space densely embedded in a Hilbert space?

Can every Banach space be densely embedded in a Hilbert space? This is clear if the Banach space is actually a Hilbert space, but much can you relax this? If the embedding exists, is the target ...
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25 views

Show that retarded argument functional operator is surjective

I want to show that for every $\lambda \neq 0$ and $g\in L^{2}(\mathbb{R})$, there is a $f \in L^{2}(\mathbb{R})$ such that $$e^{-|x+1|}f(x) - \lambda f(x+1) = g(x)$$ I tried looking at the operator ...
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1answer
24 views

Existence of $L^{2}(\mathbb{R})$ function satisfying retarded argument equation

This question arose when I tried to discuss the spectrum of a special $L^{2}(\mathbb{R})$-operator. Let $g(x) = e^{-x}$. For which $\lambda \in \mathbb{C}$ is there a function $f \in ...
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22 views

Characterization of square-summable sequences

I'm curios whether or not the following implication is true: If $x_{n} \notin \ell^2{(\mathbb{N})}$, is there necessarily a sequence $y_{n} \in \ell^{2}(\mathbb{N})$ such that $x_{n}y_{n} \notin ...
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1answer
85 views

$c_0$, the space of sequences converging $0$ is complete with dual $\ell^1(\mathbb{N})$

Let $c_0$ be the space of all complex sequences $(a_n)$ such that $$\lim_{n \to \infty} |a_n| =0$$ with norm $\|(a_n)\|_{c_0} = \sup_{n} |a_n|$. Is it fair to say that: Let $\{(a_n)\}_{n \in ...
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1answer
51 views

Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...
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1answer
26 views

$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
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23 views

the space $\mathcal{C}(\Omega),\quad \mathcal{C}^1(\Omega)$

Is the spaces $\mathcal{C}(\Omega),\quad \mathcal{C}^1(\Omega),\;\Omega \;\mbox{open set in} \;\mathbb{R}$ hilbert ? If so, how to prouve that ?
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29 views

Can one define (appropriately) a Fourier tranform on a Hilbert space?

I wonder if there is any theory or developement on Fourier transforms on geneal Hilbert spaces $H$. I know there are developements in locally compact Hausdorff spaces but $H$ is unfortunately not ...
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1answer
25 views

Left Shift Operator Spectrum Q2

Consider $\ell^2(\mathbb{Z})$. Let $R: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be such that $R((a_n)) = (a_{n+1})$. I need to prove that, given $z \in \mathbb{C}$ with $|z| >1$, the two series ...
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32 views

Can a self adjoint positive operator have a square root that is not comparable to 0?

So, I've just learned about the definition of the square root of a positive self adjoint operator in a Hilbert space, and seen the proof that it has a unique positive square root. (Which would then ...
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4answers
180 views

Left Shift Operator Spectrum

Consider the Hilbert space $\mathcal{H}=l^2(\mathbb{Z})$ and define the left shift operator $\mathcal{L}:\mathcal{H} \rightarrow \mathcal{H}$ by $$ \mathcal L (a_n) = (b_n) \qquad \text{ where } ...
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22 views

If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $|f_n|_H - |g_n|_H \to |f|_H-|g|_H$, does $f_n \to f$ and $g_n \to g$?

We work in a Hilbert space $H$. If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $$|f_n|_H - |g_n|_H \to |f|_H-|g|_H$$ is there any chance that $f_n \to f$ and $g_n \to g$? Of course this ...
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32 views

Find a function $b$ such that the operator $\frac{d}{dx}+b(x)$ is symmetric with the weight $x^2$

Find the value of $b(x) \in \mathbb{C}, x\in \mathbb{R}$, so that $$Â=(Â^{*})^{t}$$ with $$Â=i\frac{d}{dx}+b(x)$$ Here, $(f|g)$ is defined by $$ \int_{-\infty}^{\infty} x^{2}f^{*}(x)g(x)dx $$ I ...
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1answer
29 views

Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
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1answer
49 views

Antiderivative as an integral operator from $L^2(0,2\pi)$ to $L^2(0,2\pi)$

I am starting to study Functional Analysis on Hilbert Spaces and I am studying the following operator: $$T:L^2(0,2\pi) \rightarrow L^2(0,2\pi) $$ where $$Tf:(0,2\pi) \rightarrow \mathbb{R} \\ ...
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1answer
63 views

Inner product from Fourier-like kernel

A kernel $K\colon [0,1]^s\times [0,1]^s \rightarrow\mathbb{R}$ is a symmetric and positive semi-definite function (meaning that for any $v_1,\ldots,v_m\in [0,1]^s$ and any $m\geq 1$, the matrix ...
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31 views

$W = \text{Span}(V \cup\{y\})$ is a closed subspace if $V$ is. (In Hilbert Space) **Edited**

Let $V$ be a closed subspace of $\mathcal H$ and if $y \in \mathcal H$ but $y \notin V$ prove that $W = \text{Span}(V \cup\{y\})$ is a closed subspace. Here's my attempted proof. Proof: Note ...
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40 views

Topological characterization of the range of a bounded normal operator

Let $T$ be a bounded normal operator on a Hilbert space $H$. I want to prove the following statement: $\text{ran}(T)$ is closed if and only if 0 is not a limit point of $\sigma(T)$. I tried to use the ...
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1answer
12 views

An example of symmetric, coercive, discontinuous bilinear form over a Hilbert space?

Can you show me an example of symmetric, coercive and discontinuous bilinear form over a Hilbert space? I saw some stuff here Give an example of a discontinuous bilinear form. but the forms there are ...
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1answer
29 views

Determining adjoint operator between spaces with different inner products

I'm given a space of $X \in R^{mxn}$ with inner product $\langle X_1, X_2 \rangle = tr(X_1^T X_2)$ and another space of random vectors $Y \in RV^m$ with inner product $\langle y_1,y_2 \rangle = ...
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45 views

Given two Hilbert Spaces, one is Isomorphic to a subspace of the other.

I suppose I'm doing something wrong as this solutions seems too simple. We know that each of $\mathcal{H}_1$, $\mathcal{H}_2$, has a maximal orthonormal basis. Suppose $\{u_\alpha\}_{\alpha \in ...
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14 views

Orthogonal complement of a single element of Hilbert space

Let $F = \{e\}$ be a set, where $\|e\| = 1$. My question is, what is $P_{F^\perp}h$ for all $h$ in a Hilbert space? Another question is, does $e$ all by itself, make up an ortonormal system?
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3answers
56 views

Having trouble showing a subspace of $\ell^2$ is closed.

Let $M =\{ (x_n)_{n \in \mathbb{N}} \in \ell^{2} \mid x_{2j} = 0, \text{ for all } j \in \mathbb{N}\}$. $M$ is a subspace of the Hilbert space $\ell^{2}$ and I'm supposed to show it's closed. ...
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93 views

Questions about Hilbert spaces, linear subspaces and orthonormal bases

I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks ...
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17 views

The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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2answers
47 views

Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces ...