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

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Symmetricity of square-integrable functions respect to inner-products.

Let $f(x),g(x) \in \mathrm{L}_2$. If we define inner-product $(f(x),g(x)) = \int_a^b f^*g dx$, then prove that $(f,pg) = (pf,g)$, given $f(1) = Cf(0), C=e^{i\psi}, p=-i d/dx, a=0, b=1$. I tried ...
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1answer
20 views

Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
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1answer
30 views

Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.

Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...
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33 views

C*-algebra representation

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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1answer
42 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb ...
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0answers
39 views

Trace class operators problem

Let $\mathcal{B}_1(\mathcal{H})$ be the set of trace class operators in a Hilbert space $\mathcal{H}$ and $\mathcal{H}^{(d)} = \bigoplus_{i=1}^d \mathcal{H}$ with $1 \leq d \leq \infty$. If $C \in ...
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2answers
38 views

A question about reflexive spaces

Quoting wikipedia "a normed vector space is reflexive if it coincides with its bidual". Another definition, more precise is that a normed vector space is reflexive if its evaluation map ...
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1answer
64 views

Composition of two orthogonal projections

Let $V$ be a finite dimensional Euclidean space and let $W_1,W_2$ be two subspaces of $V$. Let $P_1,P_2$ denote the projections onto $W_1,W_2$ respectively. Is it true that the composition $P_1\circ ...
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1answer
42 views

What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
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1answer
62 views

Do there exist two vectors in a Hilbert space such that $(x,y)\geqslant k\|x-y\|^{-2}$?

Let $H$ be a Hilbert space, $(x,y)$ denote the inner product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$. Do there exist such $x,y\in H$ that $$ ...
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2answers
42 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
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1answer
13 views

For bounded operator $U$, show that if $UU^*$, $U^*U$ are projections, then $U$ is a partial isometry

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $U : \mathcal{H} \to\mathcal{H}$ is a bounded linear operator such that $UU^*$ and ...
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0answers
34 views

Seeing that a function is a trigonometric polynomial

I'm working through Chapter 4 of Rudin's Real and Complex Analysis book right now, and I've found myself rather more confused than usual. In the proof of the completeness of the trigonometric system, ...
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0answers
117 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
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1answer
31 views

Question about different defintions of isometry on a Hilbert space

Let $(\mathcal{H} , (\cdot, \cdot))$ be a Hilbert space over the field $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$ (so the norm on $\mathcal{H}$ is given by $\|\cdot\| = (\cdot, \cdot)^{\frac{1}{2}}$). ...
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2answers
69 views

Looking for a book: $B(H)$ not reflexive

I'm looking for a book with a proof that for an infinite dimensional Hilbert space, $B(H)$ is not reflexive. Thank you.
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1answer
44 views

No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
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2answers
51 views

ONB: Density Check?

How to show that $\{\sin{kx}:k\in\mathbb{N}\}$ for $\{f\in\mathcal{L}^2[0,\pi]:f(0)=f(\pi)=0\}$ is an ONB? (Clearly they are orthogonal to each other but is their span also dense?) What general ...
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0answers
26 views

There is a unit sequence weakly converging to every element of unit ball

Suppose the Hilbert space $H$ has a countable (I assume Hilbert?) basis. Let $x \in H$ be such that $\lvert x \rvert \leq 1$. Show that there exists a sequence $\{u_{n}\}$ in $H$ with ...
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1answer
30 views

Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces. [duplicate]

Is it true that $M=(M^{\perp})^{\perp}$ if $M$ is a closed subspace of a PRE hilbert space (a space with a scalar product, but that is not complete)? The proof of the analog fact for hilbert spaces ...
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2answers
32 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
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2answers
74 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
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1answer
92 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
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2answers
28 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
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1answer
34 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
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1answer
18 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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2answers
68 views

Minimum of $f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$

Show that $$f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$$ is defined on $\Bbb{R}^2$, it has a minimum and find for which couple $(x, y)$ the minimum is reached. The first point is okay, ...
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53 views

Are these functions on a Hilbert space Lipschitz equivalent?

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. Fix a bounded operator $T$ on $H$, and $1\leq p<\infty$ (you can assume $p$ is an integer if necessary). Consider the ...
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1answer
18 views

Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
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1answer
129 views

How to Prove the Semi-parametric Representer Theorem

This question concerns the generalized Representer Theorem, due to Schölkopf, Herbrich, and Smola. In this magnificent work, the authors provide two versions of the Representer Theorem, a ...
2
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1answer
48 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
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1answer
29 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
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1answer
28 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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15 views

H2 norm or H2 control probloem

I read a paper, it says: How to understand this definition of H2? I know the definition of the H2-norm of a system: or However, how to understand the paper one's from these formula or ...
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1answer
75 views

Momentum Operator: Selfadjoint Extensions

This might be a possible duplicate - please let me know if there is already a proof in another thread. Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$: ...
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0answers
28 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
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64 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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1answer
30 views

Approximations of compact operators

Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
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1answer
28 views

Positive-definite function on a group function on a group

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
2
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1answer
51 views

Composition of Partial Isometries

Let $H$ be a complex Hilbert space and $S,T \in B(H)$ partial isometries. Then $S T$ is a partial isometrie, if and only if $T^*(\ker(S)) \subseteq \ker(ST)$. Edit: My attempts so far: ...
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1answer
31 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
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2answers
46 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
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19 views

does the hilbert space construction of random variables allow for infinite variance?

I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner ...
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1answer
37 views

Does there always exist an equivalent geometric problem for a given differential equation?

I recently came up with a method to show for any given differential equation whose solution is the power series there exists an equivalent geometric problem in Hilbert space. I was wondering if ...
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1answer
19 views

Question about a scalar product

It is well known that in a hilbert space $H$ with orthonormal basis $(e_n)_{n=1}^{\infty }$, we have for every $f, g \in H$ $$\displaystyle{\langle f,g\rangle=\sum_{n\in\mathbb{N}}\overline{ \langle ...
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0answers
59 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$
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1answer
26 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
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1answer
28 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
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1answer
27 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
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1answer
38 views

Canonical isometric isomorphism of $l_{\alpha}^{2}$

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} ...