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

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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
47 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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2answers
190 views
+100

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
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2answers
44 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 ...
2
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1answer
34 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 ...
2
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1answer
47 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 ...
8
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1answer
126 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 ...
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1answer
41 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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2answers
50 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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0answers
24 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 ...
2
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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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50 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 $$ ...
2
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2answers
29 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 ...
3
votes
1answer
2k views

Volterra Operator is compact but has no eigenvalue

Volterra operator is defined as operator $V:L^2[0,1]\rightarrow L^2[0,1]$ by \begin{eqnarray} (V)(f(x))=\int_0^xf(y)dy \end{eqnarray} Would you help me to prove that this operator is compact but has ...
2
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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
33 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
16 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 ...
3
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1answer
69 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
2
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1answer
59 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
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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, ...
0
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1answer
17 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?
2
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0answers
52 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 ...
8
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334 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
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1answer
28 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
26 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.
7
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1answer
117 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
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0answers
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 ...
3
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0answers
63 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
71 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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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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1answer
248 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: M is a non-empty set in an Inner Product Space (IPS) X. I need to show that the annihilator of M which is ...
2
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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
49 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: ...
2
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1answer
68 views

is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$. let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim ...
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1answer
53 views

Creation and Annihilation Operators: Norm Estimate

Given the Fock space: $$\mathcal{F}(\mathcal{h}):=\bigoplus_0^\infty\mathcal{h}^{n}\text{ with } \mathcal{h}^{n}:=\bigotimes_1^n \mathcal{h},\mathcal{h}^0:=\mathbb{C}$$ Define the creation and ...
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1answer
30 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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1answer
36 views

Inner product on Hilbert Spaces

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?
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2answers
131 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
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1answer
28 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
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1answer
15 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
2
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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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2answers
87 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
2
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2answers
174 views

Net convergence and norm-convergence in Hilbert spaces

Let $\mathcal H$ be a Hilbert space which is not necessarily separable. Given a sequence of element $h_n$ indexed by $\mathbb N$, we say their sum converges in norm if the sequence $\{\sum_{n=1}^k h_n ...
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1answer
234 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
2
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0answers
18 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
36 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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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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0answers
58 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 $\sum_n|b_n|^2<\infty$