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

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183 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 ...
3
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2answers
287 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 ...
3
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2answers
585 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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1answer
101 views

Orthogonal matrix

Let $A$ be a real $n \times n$ matrix. I'm trying to prove that if $A$ maps orthogonal vectors into orthogonal vectors and $\lVert A \rVert = 1$, then, for every $x,y \in \mathcal{l}_2^n$, ...
3
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1answer
111 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 ...
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1answer
415 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 ...
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1answer
345 views

-Almost- self-adjoint bounded operators on Hilbert spaces

I ran into the following question about bounded operators on Hilbert spaces; I could really use some help. It goes like this: Suppose that $\left\{T_{k}\right\}$ is a collection of bounded operators ...
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1answer
50 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
3
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1answer
59 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
3
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1answer
73 views

Spectral resolution of multiplication operator

Kosaku YOSIDA claims in his book "Functional Analysis" that it is easy to see that the multiplication operator $Hx(t) = tx(t)$ in $L^2(-\infty,+\infty)$ admits the spectral resolution $H = ...
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1answer
30 views

Bessel potential space: Proof of completeness

I want to know a proof that the (one-dimensional) Bessel potential space (for $p=2$) $$H^s(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}:\int_{\mathbb{R}}(1+\lvert \xi\rvert^2)^{\frac{s}{2}}\lvert ...
3
votes
2answers
53 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 ...
3
votes
1answer
37 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
3
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1answer
103 views

Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a function: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad ...
3
votes
2answers
29 views

Terminology for orthogonal projections

Let $H = X \oplus Y$ a Hilbert space. Then, the map $p(x + y) = x$ is called the orthogonal projection onto $X$ along $Y$. Why is it necessary to mention along $Y$? Of course if a space has a ...
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votes
1answer
40 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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votes
1answer
93 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
3
votes
2answers
102 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
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1answer
45 views

Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
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1answer
62 views

Normal Operator: Everywhere defined implies bounded?

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{H}\to\mathcal{H}$. If its domain is the whole Hilbert space then is it necessarily bounded? The point is that I'm trying ...
3
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1answer
35 views

Family of sequences in a Hilbert space with certain property

Suppose $\mathcal{F}$ is a family of sequences on the unit sphere of $l_2$ with the following property: For any sequence $\varepsilon_n\downarrow 0$ but which is not eventually identically $0$, there ...
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1answer
63 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
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1answer
31 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
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1answer
61 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
3
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1answer
130 views

Projection operator in Hilbert space

Let $H$ be a Hilbert space, can we find an increasing net of finite rank projections which converge to the identity in the strong operator topology? And I think if $H$ is separable, we can find an ...
3
votes
2answers
186 views

Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...
3
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1answer
81 views

Sesquilinear Forms: Reals

Given a real Hilbert space $\mathcal{H}$. Consider symmetric forms: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{R}:\quad s(\psi,\varphi)=s(\varphi,\psi)$$ By polarization one obtains: ...
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1answer
86 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
3
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1answer
90 views

Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
3
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1answer
53 views

Proving $||A||=||A^{*}||=||AA^{*}||^{1/2}$

I am studying functional analysis, In the lecture notes I saw the claim: Let $A\in L(\mathcal{H})$ where $\mathcal{H}$is a Hilbert space. then $$ ||A||=||A^{*}||=||AA^{*}||^{1/2} $$ There is ...
3
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1answer
246 views

Separable Hilbert space weak sequential compactness

To preface, Banach-Alaoglu shows weak* sequential compactness of the unit ball, and in Hilbert spaces weak* and weak convergence is the same. So I already know that the unit ball of a Hilbert space is ...
3
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1answer
163 views

How is the inner product in $H^{-1/2}$ defined?

Since $H^{1/2}$ is a Hilbert space, $H^{-1/2}$ must also be a Hilbert space by the isomorphism of Riesz representation theorem. How is the inner product defined there? We know there is a nice ...
3
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1answer
129 views

Is my proof that a function is measurable correct?

Let $V$ be separable and Hilbert. Let $\mathcal V = L^2(0,T;V)$. Assume for each $t \in [0,T]$, $$a(t;\cdot,\cdot):V \times V \to \mathbb{R}$$ is continuous and bilinear. Or equivalently, we have ...
3
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1answer
110 views

The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
3
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1answer
87 views

How to prove that the set $e^{inx}$ is closed with respect to this measure?

Why is the set $\{e^{inx}\}$ closed in $L^2(d\nu)$, where $d\nu(x)=(1+|h(x)|)dx+|ds(x)|$? $d\nu$ is defined in this proof I am struggling to understand, where $\mu$ is a complex Borel measure on ...
3
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1answer
857 views

Spectral Theorem for bounded compact, self-adjoint operators as corollary of Hilbert-Schmidt theorem

I'm following Debnath and Mikusinksi's "Introduction to Hilbert Spaces with Applications" and am trying to understand how the spectral theorem for compact self-adjoint operators is a corollary of the ...
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1answer
551 views

about closed linear subspace

Can you help me, plese, with the notion of closed linear subspace. What means, examples of closed linear subspace, how can I prove that a subspace is a closed linear subspace. Thanks :-)
3
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1answer
360 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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1answer
214 views

eigenvalue question

I think this question isn't that hard, but I am a bit confused. Define the linear operator $T_k:H\mapsto H$ by \begin{align} T_ku=\sum^\infty_{n=1}\frac{1}{n^3}\langle u,e_n\rangle e_n+k\langle ...
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1answer
692 views

The relation between bounded invertible and surjective operators

Please, answer me that how is the set of all bounded invertible operators (for example on a Hilbert space) clopen (closed and open) in the set of all bounded surjective operators? In fact, which ...
3
votes
1answer
212 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 ...
3
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1answer
135 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 ...
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1answer
60 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 ...
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2answers
133 views

closedness of image of closed, unbounded operator

I want to prove the following: Suppose $\mathcal{H}_1$ and $\mathcal{H_2}$ are Hilbert spaces and let $T: \mathcal{D} \rightarrow \mathcal{H}_2$ be a closed operator, where $\mathcal{D} \subset ...
3
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1answer
422 views

Derivative of Convex Functional

Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this ...
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1answer
86 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
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1answer
127 views

Balls in the space of bounded operators on a Hilbert space

Suppose $\mathsf{H}$ is an infinite-dimensional (non-separable preferably) Hilbert space. Consider the space $L(\mathsf{H})$ of all bounded operators on it. Is there $0\neq W\in L(\mathsf{H})$ such ...
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0answers
18 views

Verification of a theorem regarding Mercer Kernel.

if $\langle x,y\rangle$ is a Mercer kernel, then is $\langle c_1 x_1 + c_2 x_2,y\rangle$ a Mercer kernel where $c_1+c_2=1$? Ans: I give the following (dirty) line of proof. Please tell whether its OK ...
3
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1answer
68 views

Normal Operators: Superalgebra (I)

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their calculus: ...
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54 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...