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

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2
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2answers
48 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...
0
votes
1answer
8 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...
1
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0answers
26 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
1
vote
1answer
5 views

Spectral Measures: Scale Spaces (V)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
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1answer
13 views

Spectral Measures: Scale Spaces (IV)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
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1answer
18 views

Spectral Measures: Scale Spaces (III)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
votes
1answer
29 views

Spectral Measures: Scale Spaces (I)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
1
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0answers
28 views

series of linear operators

Let $\mathcal{B}(\mathcal{H})$ be the Banach space of bounded linear operators on a complex, separable, infinite-dimensional Hilbert space $\mathcal{H}$. It is well known that ...
4
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1answer
41 views

Range of normal operator and its adjoint are equal

On Wikipedia it is written that bounded normal operator in Hilbert space has the same range and kernel as its adjoint. I've been able to show equality of kernels and closures of ranges: ...
2
votes
1answer
35 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
2
votes
0answers
41 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 ...
5
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1answer
41 views

Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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0answers
32 views

Show that $ℓ_2(X)$ is Hilbert space for every set $X$

Show that $ℓ_2(X)$ is Hilbert space for every set $X$ I tryed to find a proof for this problem but i couldn't (searched on internet and mathematical books.Can we find a completed proof for this?
1
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1answer
53 views

Does $A$ and $(A+I)^{-1}$ commute for positive operator $A$?

Suppose that $A$ is a bounded positive operator ($A \geqslant 0$) on some Hilbert space. Can I say that $A$ and $(A+I)^{-1}$ commute?
0
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0answers
25 views

Prove or disprove that $φ_v:u\mapsto \langle\mathcal A u,v\rangle$ is in $V^*$

Let us consider a linear and continuous operator on a Hilbert space $V$, $\mathcal A:V\rightarrow V$, such that: $$\|\mathcal A u\|\leq M \|u\|, \ \ \forall u\in V, M>0$$ and now consider ...
2
votes
1answer
39 views

Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
1
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1answer
21 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
0
votes
1answer
20 views

Infinite sum of bounded linear operators on a Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, separable, complex Hilbert space, and let $\mathbf{a}$ and $\mathbf{b}$ be bounded linear operators on $\mathcal{H}$ such that ...
0
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1answer
42 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
0
votes
0answers
21 views

Domain of closed unbounded operator

Let $A$, $B$ be two closed unbounded operators such that: (1) there exists dense subspace $\mathcal{D}$ of $Dom(B)$ which is contained in $Dom(A)$, (2) for every $\psi \in\mathcal{D}$ it holds $$ ...
1
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1answer
42 views

How to express double orthogonal complement?

Let $V$ be a Hilbert space and $U \subseteq V$. Then $U^\perp = \{\mathbf{v} \in V|\forall \mathbf{u} \in U, \langle \mathbf{u}, \mathbf{v} \rangle = 0 \}$. My question is, how do you express ...
2
votes
1answer
28 views

Existence of the continuous spectrum of a possibly-unbounded, linear self-adjoint operator on a complex Hilbert space

Let $\mathbf{A}$ be a possibly-unbounded, linear self-adjoint operator on an infinte-dimensional, complex separable Hilbert space $\mathcal{H}$, and suppose we know the matrix elements $\langle ...
0
votes
1answer
21 views

Prove that $U$ does not have closed range [closed]

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis for a Hilbert space $ H$ and define $ U$ by $ Ue_k = e_k + e_{k+1} $ . Prove that $U$ does not have closed range.
0
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1answer
15 views

What is the analogy between how logical relations are defined in set theory and hilbert space?

I am reading about hilbert spaces ( in relation to quantum mechanics ). The book I am reading ( link is not available ) tries to tell how logical relations are defined in hilbert space. I am confused ...
0
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0answers
20 views

is every n-dimensional subspace of l2 isometrically isomorphic to l2n?

Let $E$ be an $n$-dimensional subspace of $\ell_2$. I seem to recall hearing that $E$ must be isometrically isomorphic to $\ell_2^n$, but I can't see why this would be the case, nor can I find a ...
0
votes
1answer
29 views

Is $C_0^\infty(\mathbb{R}_+)$ a dense subspace of $W_0^{1,2}(\mathbb{R}_+)$?

I read that in some lecture notes that the space of $C^\infty$ funtions compactly supported on the positive real line is a dense subspace of the Sobolev space $W_0^{1,2}(\mathbb{R}_+)$. How can one ...
2
votes
1answer
49 views

Can every vector space (over $\mathbb{R}$ or $\mathbb{C}$) can be a Banach space (or Hilbert space)?

For a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$) with Hamel basis of cardinality $\kappa$ such that $\kappa^{\aleph_0} = \kappa$, can we define inner product(or norm) on $V$ such that $V$ is ...
0
votes
1answer
27 views

How to justify $\lVert \sum_{j=n+1}^\infty a_jh_j\rVert^2 \leq \sum_{j=n+1}^\infty a_j^2$ when $h_j$ are orthonormal

We work in a Hilbert space $H$. I want to show that a series $\sum_{j=1}^\infty a_jh_j$ converges where $h_j$ is an orthonormal basis of $H$. To do this, I want to show that the tail $$\lVert ...
0
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1answer
25 views

Hilbert space and uncountable cardinal

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?
5
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1answer
65 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...
2
votes
0answers
121 views

Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

Let $H$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{T : H \to H, T \space \text {is non-bounded and linear operator} \}$. We say An operator $T \in B(H)$ is chaotic if $T$ is ...
0
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2answers
23 views

property of orthonormal systems and sequences in Hilbert space

Problem: Let $H$ be a separable Hilbert space and {$e_n$} a complete orthonormal system of $H$. Prove that, if {$y_k$} is a bounded sequence in $H$, the condition $\lim_{k→∞} (e_n , y_k ) = 0$ for ...
0
votes
1answer
36 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
0
votes
1answer
18 views

Spectral Measures: Special Spectrum

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
votes
1answer
13 views

Reducing Spaces: Decompostion

This thread is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Regard a decomposition: ...
3
votes
2answers
43 views

Spanning set is closed.

Suppose $\{e_1,e_2,\ldots,e_n\}$ is an orthonormal set in $\mathscr{H}$ (Hilbert space) and define $$M \equiv \operatorname{span}\{e_1,e_2,\ldots,e_n\}.$$ Show that $M$ is closed. Can I show that ...
1
vote
1answer
28 views

Spectral Measures: Multi Version (III)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
-1
votes
1answer
28 views

Borel Measures: Coproduct

I need this thread as lemma! (See the advice: SE: Q&A) Given Borel spaces $\Omega_\lambda$. Consider the coproduct: ...
1
vote
1answer
17 views

Spectral Measures: Multi Version (II)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
1
vote
1answer
26 views

Spectral Measures: Multi Version (I)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad ...
1
vote
1answer
15 views

Proof of Hilbert Projection Theorem

If M is a closed subspace of the Hilbert space H and $x \in H$, then: There exists a unique element $\hat{x} \in M$ such that: $\|x-\hat{x} \|=\inf_{y \in M}\|x-y \|$ To proof of the existence of ...
0
votes
0answers
14 views

Does conjugation by half invertible matrices preserve spectrum?

Conjugation by an invertible matrix preserves the spectrum, but does conjugation by a left/right invertible matrix also preserve spectrum? My motivating situation was considering non-unitary ...
1
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1answer
26 views

Spectral Measures: Adjoint

This thread is only Q&A! (See the hint: SE: Q&A) Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the ...
0
votes
1answer
10 views

Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
1
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1answer
28 views

Riesz Representation Theorem in Wikipedia vs. Rudin's RCA

In Rudin's Real & Complex Analysis theorem 2.14, the Riesz representation theorem gives (in my very rough phrasing) an injection from linear functionals on a space to positive Borel measures which ...
1
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0answers
17 views

Total set in a Hilbert space

Definition: A subset of a Hilbert space is total if its span is the entire space. Halmos in his book (A Hilbert space problem book) asks below question: There exists a total set in a Hilbert ...
2
votes
0answers
21 views

Dimension of a Hilbert space

Halmos in his book (A Hilbert space problem book) says, 1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. 2- Also he proves if orthogonal dimension of ...
1
vote
1answer
24 views

Spectral Measures: Boundedness

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
0
votes
1answer
43 views

Normal Operators: Transform (III)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$N=Z\sqrt{1-Z^*Z}^{-1}$$ Especially one had: ...
0
votes
1answer
34 views

Spectral Measures: Invertibility

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...