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

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21 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 ...
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1answer
18 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 ...
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1answer
19 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 ...
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1answer
15 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
40 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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0answers
23 views

Continuously differentiable operator

if i consider the operator $A$ defined on $H^1_0$ by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s),~~t\leq s\\s(1-t),~~s\leq t\end{cases}$$ What is the expretion of $A'u$ ...
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0answers
16 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
40 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
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1answer
26 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 [on hold]

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
16 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
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
47 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
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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1answer
63 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
120 views

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

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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2answers
21 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 ...
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1answer
35 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
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: ...
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1answer
12 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
41 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
25 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
26 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
16 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
22 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
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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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 ...
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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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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: ...
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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 ...
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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 ...
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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 ...
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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: ...
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1answer
37 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: ...
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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: ...
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1answer
24 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
45 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
48 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
31 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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1answer
25 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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4answers
59 views

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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2answers
33 views

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
30 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
50 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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36 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 ...
4
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1answer
61 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
15 views

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
29 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 ...