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

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Mourre Adjoint: Bounded Maps (I)

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: ...
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1answer
49 views

Mourre Adjoint: Bounded Maps (III)

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: ...
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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 ...
3
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2answers
51 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 ...
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1answer
43 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 ...
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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 ...
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1answer
43 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 ...
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0answers
22 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 $$ ...
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1answer
28 views

Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
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2answers
112 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
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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 ...
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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.
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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 ...
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0answers
39 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
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1answer
50 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 ...
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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 ...
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1answer
62 views

Reducing Spaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote for readability: $$\mathcal{D}:=\mathcal{D}(N)=\mathcal{D}(N^*)$$ ...
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
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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 ...
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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 ...
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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 ...
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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?
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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
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1answer
50 views

Normal Operators: Transform (II)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ By the previous ...
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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 ...
0
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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, ...
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1answer
64 views

Spectral Measures: Domain Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Then the criterion holds: ...
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2answers
62 views

Spectral Measures: Unitary Map [duplicate]

This thread is a record. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}\to\mathcal{H}:\quad N^*N=NN^*$$ and its spectral measure: ...
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1answer
56 views

Spectral Measures: Embedding

This thread is just a note! 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: ...
2
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2answers
67 views

Spectral Measures: Support

Given and a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Define its support: ...
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2answers
48 views

Spectral Measures: Norm

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: ...
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3answers
66 views

Spectral Measures: Support vs. 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
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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: ...
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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: ...
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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
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1answer
19 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: ...
9
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1answer
134 views

Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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3answers
185 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
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1answer
38 views

Reducing Spaces: Characterization

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\mathcal{D}:=\mathcal{D}(T)$$ Regard a subspace: ...
0
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1answer
35 views

Reducing Spaces: Preliminary

Given a Hilbert space $\mathcal{H}$. Consider a dense domain: $$\mathcal{D}\leq\mathcal{H}:\quad\overline{\mathcal{D}}=\mathcal{H}$$ Regard a closed subspace: ...
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1answer
30 views

Reducing Spaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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1answer
52 views

Reducing Spaces: Complement

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\overline{\mathcal{D}(T)}=\mathcal{H}$$ Regard a subspace: ...
3
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1answer
354 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
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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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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 ...
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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 ...
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0answers
15 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 ...
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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: ...
0
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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
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: ...