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

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98 views

Differential Operator on $L_{2}$ problem

I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated, Consider the partial differential equation, $\frac{\partial ...
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0answers
84 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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1answer
93 views

Is this function positive?

I was wondering if: $$\int_0^1x(t)\int_0^tx(s)ds\ dt$$ is positive for a general $x\in L_2[0,1]$ . Can you help me with this?
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1answer
65 views

Finding the minimizing vector of a $l_{2}$ sequence

I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated. Let $X = l_{2}$. Let $x \in X$ be given by $x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$ Let $M ...
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1answer
156 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...
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488 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} ...
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1answer
342 views

cyclic vector exists for symmetric operator iff there no repeated eigenvalues

Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
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2answers
137 views

Conclusion in the proof that every Hilbert space has an orthonormal basis regarding countable index set

I am working through a proof that every hilbert space has a orthogonal basis which lies dense in that Hilbert space. In the proof the following is done: Let $v$ be a vector, and $E \subseteq I$ an ...
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1answer
80 views

How do I prove that a particular linear operator has an orthonormal basis?

I have to show that if $T$ is a linear operator such that $T: L^2(\mathbb(R)^n) \to L^2(\mathbb(R)^n)$ and $T(f)(x) = \int_{R^n}f(y)g(x,y)dy$, where $g(x,y)$ is an $L^2$ function, that there is an ...
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1answer
227 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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1answer
76 views

Approximating bounded operators in Hilbert space

Let $H$ be a separable Hilbert space, show that every bounded operator from H to itself can be approximated in the strong operator topology by a sequence of finite rank operators. I know we can find ...
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3answers
1k views

Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$. Let $X$ be a Hilbert space, would you help ...
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1answer
272 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
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1answer
123 views

Symmetric bounded linear maps can be approximated by compact symmetric linear maps.

Let $H$ be a separable Hilbert space and let $T:H \rightarrow H$ be a symmetric bound linear map. a) Show that for every orthogonal projection $P$ on $H$ ($P' = P$, $P^2 = P$) PTP is symmetric. b) ...
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2answers
919 views

Compact set in a Hilbert Space

I'm doing some exercises, but there is something in one of the questions I don't quite understand, so I'm really hoping someone might be able to clarify. The question is as follows: Let $\mathcal H$ ...
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2answers
253 views

Weakly convergent sequence

Consider a sequence $(x_n)_n$ in Hilbert space $H$ such that $\langle x_m,x_n\rangle=\delta_{mn}$ where $\delta_{mn}$ equals one if $m = n$ and $C$ otherwise. Prove that $(x_n)_n$ is a weakly ...
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1answer
207 views

Normal operators in Hilbert spaces

Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
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1answer
131 views

Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
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2answers
631 views

Spectrum of an Orthogonal Projection Operator

I want to show that $ \sigma(p) = \{ 0,1 \} $ for any orthogonal projection operator $ p \notin \{ 0,I \} $ on a Hilbert space $ \mathcal{H} $. Recall that an orthogonal projection operator $ p $ on $ ...
3
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1answer
176 views

Determine the operator T in a Hilbert space

Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$. a) Determine the operator $T\in B(H)$ that satisfies $$ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n ...
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1answer
915 views

How to prove this integral operator is compact?

$T_kf=\int K(x,y)f(y)dy$ where $K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}}$ $\phi(x)$ is a smooth function on a compact support. $f$ is defined on $R^n$ and $K$ is defined on $R^n\times R^n$ ...
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2answers
103 views

If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.

This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following $\mathcal{A}$ is a Banach *-algebra. ...
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265 views

Span of functions dense in $L^2$

This is an exercise from Werner's Funktionalanalysis. I have to show that the linear span of the functions $f_n(x)=x^ne^{-x^2/2}, n\geq0$ is dense in $L^2(\mathbb{R})$. The book gives the hint to ...
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74 views

Positive polynomial in positive operator yields a positive operator?

Let $H$ be an Hilbert space, and let $T$ be a self-adjoint, positive (and therefore bounded) operator $H \to H$, with $||T||<2012$. Let $P$ be a polynomial with real coefficients such that the ...
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1answer
773 views

A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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2answers
539 views

Hilbert Adjoint Operator from Riesz Representation Theorem - $T^{*}y=\frac{\left\langle y,Tx\right\rangle }{\left\langle z_{0},z_{0}\right\rangle}z_0$

Kreyszig's Functional analysis seems to introduce the hilbert adjoint operator by means of an explicit representation. I haven't seen this anywhere else and I would like to confirm this explicit ...
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3answers
1k views

Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence ...
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1answer
115 views

Inequality of bounded linear operators on Hilbert space

Let $T$ and $S$ be bounded linear operators on a Hilbert space $H$. Verify that: $||TS||\leq ||T||\cdot ||S||$. The definition of the operator's norm is $||T||=\sup\{||Tv||_H: ||v||_H=1\}$. ...
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442 views

Proof of normal operator and self-adjoint operator

1) Let $T∈L(V,V)$ be a normal operator. Prove that $||T(v)||=||T^*(v)||$ for every $v∈V$. ($T^*$ is the adjoint of $T$) 2) Let $T$ be an operator on the finite dimensional inner product space ...
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1answer
362 views

Complex conjugate of the Hilbert space

Consider a Hilbert space $H=L^2(\mathbb{R}_+)$, take its conjugate $\overline{H} := \left\{f^{+}, f \in H \right\}$, where $+$ stands for the conjugation. Space $\overline{H}$ is a Hilbert space with ...
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0answers
169 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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1answer
83 views

Is it ok to switch the limits in $L_2$?

Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$. Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there ...
4
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1answer
323 views

Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space

Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space? The other direction is clear by Riesz' ...
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1answer
167 views

Show $T$ is compact

$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges. $T\colon H\rightarrow K$ is defined by ...
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1answer
161 views

Inequality - tensor product, Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{K}$ be a Hilbert space with an orthonormal basis $\{ e_i \}_{i \in I}$. Let $A$ be bounded linear operator from $\mathcal{H} \otimes \mathcal{K}$ ...
2
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2answers
299 views

Completion of pre-Hilbert space in H. Brezis' Functional Analysis

I'm trying to solve the problem 5.12 of Harim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations; but I'm stucked understanding the statement which comes as follows: ...
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2answers
96 views

pre hilbert space

i have got a question about pre hilbert space. In the lecture, we said that a vector space E with fixed inner product is called pre hilbert space. My question is, what does fixed mean here? Does this ...
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1answer
61 views

Norm of two operators

(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$ For this one I tried the ...
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3answers
340 views

Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
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1answer
164 views

Inverse of trace class operator restricted to it's range

A paper I'm reading constructs the Cameron-Martin space in a way different than I'm used to, and in the process they gloss over a functional analysis result about the existence of an inverse. It ...
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1answer
139 views

Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
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1answer
220 views

Countable family of Hilbert spaces is complete

Let $H_1, H_2, \ldots, H_n$ be a countable family of Hilbert spaces. Let H be the set of tuples $x = (x_1, \ldots, x_n,\ldots)\in \prod_n H_n$ with the property that $$\|x \| ^2 =\sum_n \| x_n \| ...
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3answers
845 views

Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional

Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$. I have to show that $T$ is compact iff $M$ is finite ...
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1answer
79 views

On operator norm of Hilbert spaces

I was going through some lecture notes on operators on Hilbert space and on a proof it is stated note that $||T||=\sup_{||x||=||y||=1}\text{Re}\langle Tx,y\rangle$ However I cannot see why this ...
4
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1answer
831 views

Hilbert-Schmidt Operator

We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition: If $H$ is a Hilbert space and ...
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1answer
89 views

Weighted Hilbert space

I know that the Sobolev space $H^s$ with the inner product given by $$ \langle u,v \rangle_{H^s} := \sum_{| \alpha| \leqslant s} \int_{\Bbb R^n} \nabla^\alpha u \cdot \nabla^\alpha v $$ is Hilbert ...
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1answer
244 views

Operation in Hilbert space with matrices

Let $\{e_n\mid n \in \mathbb{N}\}$ be an orthonormal basis for the Hilbert space $H$ and define for each $T \in B(H)$ the doubly infinite matrix $A = \{\alpha_{n,m}\}$ by letting $\alpha_{n,m} = (T ...
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1answer
349 views

A problem about the weak convergence in a Hilbert space.

Let $H$ be a Hilbert space and assume that $u_n$ converges weakly to $u$ in $H$. Then I want to prove that $$ \lim_{n \to \infty} \| u_n - u \|_H = 0 \;\;\Longleftrightarrow \;\; \| u \|_H \geqslant ...
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1answer
654 views

Multiplication operator

Let $M_{\phi}$ be a multiplication operator $M_{\phi}:L^{2}\left(\mu\right)\rightarrow L^{2}\left(\mu\right)$ defined by $M_{\phi}f=\phi f$. Show that $\ker M_{\phi}=0$ if and only if ...
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1answer
50 views

Inequality involving a sequence in Hilbert space

Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $$\lVert y_n \rVert_H^2 = \int_0^T ...