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

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Continuous Sobolev Embedding

Does Sobolev spaces $H^s$ continuously embed into $L^2$? It seems like this is the case from this post https://en.wikipedia.org/wiki/Rigged_Hilbert_space where can i find a list of continuous ...
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28 views

Obstruction map for Quot schemes is surjective

I am reading "Lectures on vector bundles" by Le Potier and am confused about a statement in the proof of the existence theorem on page 144, after Lemma 8.6.6. Let $X$ be a projective curve (can ...
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29 views

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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28 views

Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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36 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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19 views

Wave Operators: Cook

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...
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33 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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1answer
59 views

Singular Spectrum: Criterion

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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37 views

Normal Operators: Superalgebra

Problem Given a Hilbert space $\mathcal{H}$. Consider dense operators: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{H}:\quad\overline{\mathcal{D}A}=\mathcal{H}$$ Denote the normals: ...
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Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
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2answers
62 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 ...
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20 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 ...
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1answer
65 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 ...
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1answer
15 views

Spectral Measures: Scale Forms

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: ...
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38 views

Spectral Measures: Scale Operators

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: ...
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1answer
48 views

Spectral Measures: Scale Spaces

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: ...
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31 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 ...
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49 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: ...
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1answer
39 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 ...
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47 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 ...
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73 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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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?
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61 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?
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26 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
77 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
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 ...
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1answer
23 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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44 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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23 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
44 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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1answer
31 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
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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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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31 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
54 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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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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68 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 ...
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127 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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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 ...
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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
28 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
15 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
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 ...
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32 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
30 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
22 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
32 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
16 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 ...