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

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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 ...
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1answer
42 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
31 views

Spectral Measures: Special Spectrum

Problem Given a Hilbert space $\mathcal{H}$. Denote eigenvalues by: $$\sigma_0(N):=\{\lambda\in\mathbb{C}:\mathcal{N}(\lambda-N)\neq(0)\}$$ Then arbitrary sets admit: ...
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1answer
19 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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2answers
46 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 ...
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1answer
39 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: ...
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1answer
38 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
29 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: ...
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1answer
37 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 ...
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1answer
30 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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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
29 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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1answer
15 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: ...
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1answer
56 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 ...
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0answers
89 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 ...
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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 ...
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1answer
26 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: ...
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1answer
76 views

Spectral Measures: Existence

Problem 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: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad ...
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1answer
35 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: ...
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1answer
28 views

Why can we consider elements of a normed space $X$ as elements of a normed space $Y$, if there is an embedding between these spaces?

Let $(X,\left|\;\cdot\;\right|)$ and $(Y,\left\|\;\cdot\;\right\|)$ be normed spaces and $\iota :X\hookrightarrow Y$ be an embedding. Often when I read that such an embedding $\iota$ exists, I read ...
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1answer
62 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
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1answer
52 views

Mourre Adjoint: Approximation

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
34 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
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55 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
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Is every projection on a Hilbert space orthogonal?

I'm highly doubtful that the answer is "yes," but I fail to see what's incorrect about this very basic proof I've thought of. If someone could point out my error, I'd appreciate it. My logic is as ...
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Prove uniform convergence of $\sum_{e \in \xi} \langle Th, e \rangle e$ for $\| h \| \leq 1$

Let $\xi$ be a basis for Hilbert space $H$. From Parseval's Identity, for every $x \in H$ we have $x = \sum_{e \in \xi} \langle x, e \rangle e$. Thus, for every bounded operator $T : H \rightarrow H$ ...
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1answer
31 views

If $AT = TA$ for every continuous compact operator $T$, then $A$ is a multiple of identity

Given a Hilbert space $H$, let $A: H \rightarrow H$ be a bounded operator. Show that if $AT = TA$ for every continuous compact operator $T : H \rightarrow H$, then $A$ is a multiple of identity ...
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0answers
60 views

Show that every continuous finite rank operator $T$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$

Can someone help me with this question? Suppose that $H$ and $K$ are Hilbert spaces. Show that every operator $T \in B_{00}(H, K)$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$, where ...
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0answers
58 views

Prove that $\sin(n\pi x)$ weakly converges to $0$ in $L^2(0,1)$ [duplicate]

Let $$f_n(x):=\sin(n\pi x)\;\;\;\text{for }x\in (0,1)$$ and $$\langle f,g\rangle:=\int_{(0,1)}fg\;d\lambda^1\;\;\;\text{for }f,g\in L^2(0,1)$$ I want to show, that $(f_n)_{n\in\mathbb{N}}$ weakly ...
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1answer
89 views

Show that $\{ x_n \} \overset{T}{\mapsto} \{ \sum_{k=1}^{\infty} a_{nk} x_k \}$ is compact

Can someone help me with this question? Let $\ell^2$ be the space of complex sequences $\{ x_1, x_2, \ldots \}$ that $\sum_{n=1}^{\infty} \lvert x_n \rvert ^2 < \infty$. If $\mu$ be Counting ...
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0answers
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Closed subspace of weighted L2 space?

Let $L_{w_{\xi}}^{2}[0,\infty)$ be a weighted $L^2$-space with weight function $w_{\xi}(x) = \frac{\exp\left({-(x+\xi)^3}\right)}{(x + \xi)^2},\; \xi > 0$ and let $T$ denote the operator that ...
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1answer
35 views

A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
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1answer
44 views

Mourre Adjoint: Regularity

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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2answers
56 views

$||f||_1 =(\int_a^b [|f|^2+|f'|^2]dx)^{1/2}$. Is this normed space complete?

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$||f||_1 =\left(\int_a^b \left(|f|^2+|f'|^2\right) dx \right)^{1/2}$$ Is this normed space ...
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1answer
55 views

Mourre Adjoint: Algebra

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
31 views

Proof of positive definiteness

$Lu = -u'' + c u$ where c is some constant The question is when it's positive definite in square integrable on $[0; 1]$ with $u(0)=u(1)=0$ $(Lu, u) = \int^1_0 u Lu dx = -u u''+c u^2 dx = \int^1_0 ...
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1answer
91 views

Is $\operatorname{span}\varepsilon=\overline{\operatorname{span}\varepsilon}$ in Hilbert Space?

The term "span" is defined in linear span. So $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbf{K}} \right \}$, and ...
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1answer
30 views

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$.

Show that if $(e_n)$ is an orthonormal set in a Hilbert space $H$, the set of all vectors of the form $x=\sum c_ne_n$ is a subspace of $H$. Hint: Take a Cauchy sequences $(x_r)$, where $x_r=\sum ...
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2answers
112 views

Strong convergence from weak convergence

I am trying to show that a sequence $(x_n)_n \subseteq \mathcal{H}$ converges strongly to $x$ if it converges weakly to $x \in \mathcal{H}$ and $\|x_n\| \to \|x\|$ as $n \to \infty$ $\mathcal{H}$ is ...
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2answers
135 views

Is the unit sphere in an infinite dimensional Hilbert space closed?

Is a unit sphere in an infinite dimensional hilbert space closed. By the triangle inequality it is clear that the all the limit points of the sphere are inside the closed unit ball. But I cannot ...
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Why do dagger categories supposedly capture the structure of a Hilbert space?

A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ...
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1answer
49 views

Spectral Measures: Constructions

Any constructions welcome!!! Given a Hilbert space $\mathcal{H}$. Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ That are additive: ...
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1answer
52 views

Compatibility of topologies and metrics on the Hilbert cube

Consider the Hilbert cube $Y = [0,1]^\mathbb{N}$. It is easy to define four classes of metrics on $Y$ for $\gamma>0$ and $\omega>1$: $$d^\gamma_{sup,pol}(x,y) = \sup_{k\geq 1} ...
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1answer
107 views

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$.

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$. I know that the orthogonal complement of $X$ is the set ...
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2answers
39 views

Spectrum of double infinite shift using isometry to Fourier series

I'm trying to find the spectrum of the operator $T: l^2(\mathbb{Z}) \to l^2(\mathbb{Z})$ given by right shift but I am having some difficulties. I can show that $l^2$ is isomorphic to ...
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2answers
35 views

To show $I+ A$ is non singular

$A$ is a positive operator on Hilbert space $H$, I have to show the title of this question. Since $A $ is positive so all eigenvalues are $\ge 0$, so eigenvalues of $I+A$ are $\ge 1$, so $\det(I+A) ...
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1answer
93 views

Riesz-Fischer theorem

The aim of this exercise is to prove the Riesz-Fischer theorem for Hilbert spaces that aren't separable. Let $I$ an index set and $1\leq p \leq \infty$. Let $\mathcal{F}=\{F\subset I: F$ is ...
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1answer
35 views

Given any countable collection of non-zero vectors in a Hilbert space

Let $\{\alpha_i\}$ be a countable collection of non-zero vectors in a Hilbert space $H$. Is there exist a vector $\beta \in H$ such that $\langle \beta , \alpha_i \rangle \neq 0$ for all $i$ ?
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25 views

Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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2answers
51 views

Every non-compact Hermitian operator P has an infinite dimensional invariant subspace on which P is bounded from below

I want an explanation of the following statement. If $P$ is a Hermitian operator on Hilbert space and not compact, there exists an infinite-dimensional subspace $M$, invariant under $P$, on which $P$ ...