# Questions tagged [hilbert-spaces]

A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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### Norm of an operator on $L^2(\mathbb{R})$ [closed]

I have the following operator $A$ on $L^2(\mathbb{R}$): given $\psi(x) \in L^2(\mathbb{R}$),
$A\psi(x) = e^{-x^2}\psi(x)$. What is the norm of this operator?
By definition it should be $||A ||^2 = \...

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### Aupetit B, A primer on spectral theory, Exercise IV.1 [closed]

This is my question in
https://math.stackexchange.com/questions/3701351/aupetit-b-a-primer-on-spectral-theory-exercise-
iv-1
I'm sorry reply here
Aupetit B, A primer on spectral theory, Exercise IV....

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### Compute Frobenius inner product of two tensor-trains in terms of tensor contractions

Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...

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### Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?

Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(...

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### Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality:
$$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$
which can be found in many places in the ...

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### Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?

Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$.
Is it true that $F$ is completely contractive? If it is, I would be very grateful ...

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### How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...

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117 views

### Spherical harmonics expansion

In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics:
$$
f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{...

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### Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...

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### Covering number for the unit ball in a reproducing kernel Hilbert space

I am looking for a reference for an upper bound on the covering number for the unit ball $\{ f \in \mathcal{H}: ||f||_{\mathcal{H}} || \leq 1\} $, where $\mathcal{H}$ is a reproducing kernel Hilbert ...

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265 views

### A question about comparison of positive self-adjoint operators

I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...

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### Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]

Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...

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74 views

### Perfect images of complete Erdős space

Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact ...

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### Dense stratification of a separable Hilbert space

Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...

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### Higher-order inner products of an orthonormal basis

Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...

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### Is this subspace of $B(\mathcal{H})$ known?

Let $\mathcal{H}$ be a Hilbert space. Suppose that I take a fixed ONB of $\mathcal{H}$ let us call it $\{ e_i \}_{i\in \mathbb{N}}$ and then I define
\begin{align*}
\|T \|_{D} = \sup_{l_i, m_i} \sum_{...

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121 views

### Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]

Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$.
My question is as follows. Is there an (...

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77 views

### Mean squared absolute value of inner product of unit vectors

Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be ...

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### Continuity of local spectral radius

Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...

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156 views

### Result of continuum tensor product of Hilbert spaces

Let's suppose that with number $\mu_1 \in \mathbb{R}$ we associate a Hilbert space $\mathcal{H}_{\mu_1}$ with countable basis $|1\rangle _{\mu_1}$, $|2\rangle _{\mu_1}$, $|3\rangle _{\mu_1}$, $\ldots$ ...

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### Convergence of a succession obtained by the Gram–Schmidt process

Let $H$ be an Hilbert space over $\mathbb{C}$.
Let $\{h_n\}_{n \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ such that $h_n \to h \neq 0$ in norm topology.
We apply ...

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176 views

### Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...

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### Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...

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159 views

### Positive-semidefiniteness of a Gram-like matrix

Let $x_0, x_1, \ldots x_{n-1}$ be arbitrary vectors in a complex Hilbert space. Define the $n \times n$ symmetric real matrix $M$ by $M_{ij} = \lvert \langle x_i, x_j \rangle \rvert^2$. Must $M$ be ...

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155 views

### Sequence of Hilbert Schmidt operators

Consider the Banach space $\mathcal K=S_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$. I am looking for an example of two pairs of sequences $\{T^{(i)}\},\{\tilde T^{(j)}\}$ and $\{S^{(i)}\...

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### If $\|S\|<\sin\frac{\pi}{2n}$ then $\|P(I-S)^ku\|\neq 0$ for all $k=0,\ldots,n$

I want to show the following:
Let $H$ be a Hilbert space and let $S:H\to H$ be a bounded operator such that
$$\|S\|<\sin\frac{\pi}{2n}.$$
Let $\mathcal{L}$ be a closed subspace of $H$ and $$u_k:=...

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### Embedding random variables in infinite-dimensional spaces

Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...

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### Unboundedness of pseudo-inverse

Let $H$ be a real Hilbert space, and let $T$ be a bounded self-adjoint operator of $H$ to itself. Let $T^+$ denote the pseudo-inverse operator of $T$, that is the operator defined to be zero on $(TH)^{...

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### Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...

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### Eigenvalues and spectrum of the adjoint

In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...

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### Is there a reasonable notion of spectral theorem on a pre-Hilbert space?

I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...

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### Metric projection on CAT(0) tangent cone

Let $(Y,d)$ be a complete and separable CAT(0) space, fix $y \in Y$. Then, consider the tanget cone $(T_yY,d_y)$ at $y$, i.e. the metric cone over the space of directions, and denote by $0_y$ the 'tip'...

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### Canonical embedding of Hilbert space in random $L^2$

This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...

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### Estimate the metric entropy of unit ball in $L^2$ space

Let me clarify the setting I'm thinking.
For any totally bounded metric space $(Y,d_Y)$ and $\varepsilon>0$, the $\textit{metric entropy}$ $N_M(\varepsilon,Y)$ is the smallest number of closed ...

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133 views

### Canonical embedding of Hilbert space in $L^2$ space

Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-...

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### Numerical solution of a nonlinear saddle point problem in a Hilbert space

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be positive probability densities on $(E,\mathcal E,\lambda)$ for $i\in I$, $\mu:=p\...

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### A basic question about the Spectral Theorem

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...

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### Do all unitary representations weakly converge to zero at infinity?

Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...

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### What is the difference between $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$? [closed]

$L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$ are isometrically isomorphic because both are infinite-dimensional separable Hilbert spaces.
If a Hilbert space $H$ is $L^2(\mathbb{R})$ or $L^2(\mathbb{R}^2)...

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### A holomorphic map into a Hilbert space with prescribed orthogonality

This is a variation of my previous question.
Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...

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### Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...

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### A map into a Hilbert space with prescribed orthogonality

Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...

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### Solutions to holonomic $D$-modules: when are they square-integrable?

I want to apply the theory of $D$-modules to solve operator equations of several variables in the Bargmann space
$$\mathcal H :=\bigg\{\psi \in \mathcal O^\text{an}_{\mathbb{C}^n}\,\,\bigg|\,\,\,\...

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### What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?

Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...

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256 views

### Iterated limits equal?

Consider the Banach algebra $B_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$ with Hilbert Schmidt norm. We know that $B_2(H)$ is a Hilbert space as well with $\left<A,B\right>=tr(B^*...

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### Is it possible that the dimension of the intersection of a nested sequence of Hilbert space is 1?

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$
Let $\{h_n\}_{n \in \mathbb{N}} \in H$ be a sequence of linearly independent vectors in $H$
Let
$$
V=
\bigcap_{n=1}^\...

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### Concentration of measure on finite powers of $S^\infty$

I am wondering about a natural generalization of theorem 1.4 in the article Dvoretzky's theorem — Thirty years later by Milman. My first thought was to look at Milman's paper that he cites for the ...

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201 views

### Spectral theory in non-separable Hilbert Spaces

I am wondering about what can be said about the spectral theorem for unbounded, self-adjoint operators in a non-separable Hilbert space. There is a comment in this sense to the question "Does spectral ...

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### Weak convergence of Hilbert Schmidt operators

So I am stuck at this situation. Let $\{A_n\}$ be a weakly convergent sequence in $B_2(H)$ converging to $0$ in the weak topology on $B_2(H)$. Which means that $\left<A_n,D\right>=\operatorname{...

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71 views

### Classically compute ahead time if Lie Algebra is either polynomial finite or geometrically closed?

Given a set of $N$ operators $\mathcal{O}$ with a known set of Lie Algebra group multiplication rules $\mathcal{G}$ that can be programmed into a classical computer, is there a classical poly($N$) ...