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

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18 views

Prove that conditions are equivalent

$ X $ is unitary space, $ x,y \in X $. Prove that following conditions are equivalent: $ x \perp y $ $ ||x|| \leq ||x+ty|| $ $ t \in C $ $ ||x+ty||=||x-ty|| $ $ t \in C $ Unfortunatelly, I'm ...
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1answer
23 views

Bergman space norm in terms of coefficients

I am interested in the Bergman space $A^2$ on the unit disc. According to the Wikipedia article on Bergman spaces, if we have $f(z)= \sum_{n=0}^\infty a_n z^n \in A^2$ then $$\|f\|^2_{A^2} := ...
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0answers
25 views

Compactness in Hilbert spaces

Let $H$ be a Hilbert space with orthonormal basis $\{h_n:n\in \Bbb N\}$. Let $P_n$ be the orthogonal projection to $\operatorname{span}\{h_1,\cdots, h_n\}$. Claim: A bounded subset $U\subset H$ is ...
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2answers
26 views

Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
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46 views

Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x ...
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18 views

Show $L$ is a closed linear subspace of $H$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $<Px,y>=<x,Py>$ for all $x,y \in H$ and $P^2=P$. We can use the fact that $Px \perp (x-Px)$ for every $x \in H$ ...
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2answers
35 views

Invariant subspaces for this linear extension of operators

Let $(e_k)$ be a total orthonormal sequence in a separable Hilbert space $H$ and let $ T: H\to H$ be defined at $e_k$ by $T(e_k)=e_{k+1}$ , $(k=1,2,\cdots)$ and then linearly and ...
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3answers
44 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
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1answer
87 views

Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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1answer
18 views

Hilbert space, functional analysis

Let $X$ and $Y$ be closed subspaces of a Hilbert space $H$. Assume that dim $X < \infty$, and dim $X$ < dim $Y$. Show that $X^{\perp} \cap Y \neq \{0\}$. I want to proof it by contradiction. ...
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2answers
61 views

Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
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1answer
38 views

Eigenvectors Operators and Unilateral Shifts

We showed that a (non-zero) compact self-adjoint operator on a Hilbert space always has an eigenvector. Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator ...
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3answers
53 views

Vectors in a Hilbert space are countably supported with respect to any orthonormal basis

Let $\{e_i\}_{i\in I} \subset \mathcal{H}$ be an orthonormal set in the Hilbert space $\mathcal{H}$. For any vector $x\in \mathcal{H},$ let $$I_x=\{i\in I|\,\langle x,e_i\rangle \neq 0\}.$$ How can we ...
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34 views

Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
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0answers
18 views

Approximating step functions by Haar wavelets

Let $\psi = \chi_{[0,1/2)} - \chi_{[1/2,1)}$, then $\psi_{n,k}(t) = 2^{n/2}\psi(2^nt-k)$ with $n \in \mathbb{N}$ and $k \in \{0,1,\dots,2^n-1\}$ defines the Haar-Wavelets on $L^2(0,1)$. Let $S$ be the ...
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1answer
21 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in ...
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2answers
45 views

Question on operator theory classes of operators on Hilbert spaces

I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces: We are to find and prove the inclusion relations between the classes of operators: ...
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1answer
31 views

Norm estimation of a function on Hilbert space implies linear transformation has closed range

I want to prove the following fact: Let $T: X\to Y$ be a bounded linear transformation between two Hilbert spaces $X$ and $Y$. Show that if there exists a constant $C$ such that $\|f\| \leq C ...
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1answer
30 views

Exponential of a self-adjoint operator

Let $\mathcal{H}$ be an Hilbert space. Firstly, I shall define some notions as their definitions may vary: A spectral resolution is a function $E:\mathbb{R}\to\mathcal{L}(\mathcal{H})$ (the space ...
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38 views

orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
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18 views

Are quantum operators associative?

Let H be the Hamiltonian representing the total energy of the potential and kinetic component. But because all classical dynamical variables can be written as a function of position, x, and momentum, ...
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1answer
61 views

Sobolev spaces of sections of vector bundles

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset ...
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1answer
41 views

Completeness of derivatives of Hilbert basis with respect to a parameter

Let us take a Hilbert basis $\left|x_\lambda\right >$ in a Hilbert space $\mathcal{H}$, i.e. the $\left|x_\lambda\right >$ are a complete, orthonormal set of vectors. The subscript indicates ...
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1answer
34 views

Characteristic functions of infinite dimensional random elements

I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that ...
5
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1answer
111 views

Explicit characterization of dual of $H^1$

Let's start by some well-known facts: $H^1(\mathbb{R})$ is a Hilbert space, hence there holds the Riesz representation theorem, stating that any linear functional on it can be represented as $L = ...
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1answer
43 views

Property of bounded linear transformation between Hilbert spaces

I've asked a question on related question in a previous thread, but I wanted to ask a follow up question. If a bounded linear transformation $T: X \to Y$ where $X$ and $Y$ are Hilbert spaces has ...
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1answer
20 views

Completeness, spanning and orthonormal bases

I am having some difficulty in understanding some concepts regarding Hilbert spaces. I am learning wavelet theory (with regards to signal processing) and am reading up some basics on signal ...
2
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2answers
72 views

Proving a variant of closed range theorem on Hilbert space

I've been working on closed range theorem. There are a lot of materials on general Banach spaces, but not much on Hilbert spaces, so I was wondering if I could get some help. I'm trying to prove the ...
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35 views

Under what conditions is the resolvent set of a linear operator connected?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$. As usual, we define ...
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24 views

Show that $l^2(\Bbb N , F)$ with equipped norm is not complete.

For $v$ in $l^2(\Bbb N , F)$ with norm on $l^2$ defined as: $\lvert\lvert v\rvert\rvert_{W}= \sum^\infty_{k=1}\frac{\lvert v_{[k]}\rvert}{2^k}$ Show that $l^2(\Bbb N , F)$ with the norm ...
2
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1answer
68 views

Every partially defined isometry can be extended to a isometry

I know that the following theorem holds true: Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual ...
4
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1answer
56 views

Understanding the definition of the covariance operator

Let $\mathbb H$ be an arbitrary separable Hilbert space. The covariance operator $C:\mathbb H\to\mathbb H$ between two $\mathbb H$-valued zero mean random elements $X$ and $Y$ with $\operatorname ...
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24 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
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37 views

A question about Hilbert Spaces and convex sets

I am struggling with this and could really do with some help: Let $H$ be a Hilbert space over $\mathbb{R}$, $\{v_n\}$ be a sequence of vectors in $H$, and $C$ be a convex subset of $H$ containing ...
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4answers
1k views

What really is ''orthogonality''?

I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real ...
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24 views

Hilbert space and traces

Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define ...
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1answer
36 views

Homogeneous and Inhomogeneous Function Spaces

I would like a general explanation on the difference between homogenous and inhomogeneous function spaces, there doesn't seem to be a very good explanation online. I know that for Sobolev spaces for ...
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Computing orthogonal projection of function

Consider the Hilbert space $\ V=P^5(-1,1)$ endowed with the $ L^2 $-inner product, and the subspace $ W=P^3(-1,1) \subset V $. Let $ B_w=\left \{L_0, L_1, L_2, L_3\right \} $, where $ L_i $ are ...
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3answers
45 views

Is faithful positive sesqulinear form an inner product?

As in the title: does a positive sesqulinear form need to be conjugate-symmetric? Background: The question comes from an attempt to understand the proof of the Stinespring representation theorem. It ...
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1answer
28 views

Norm of a self adjoint operator

Let $T$ be a (bounded) self-adjoint operator on a Hilbert space. Is it true that $||T^k|| = ||T||^k$ for all positive integers $k$? It's true for $k=1,2$, and I'm wondering if this could be ...
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2answers
35 views

Why can't a Hilbert curve be used to put the real numbers into a listable format?

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ...
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1answer
179 views

When is a function of the largest eigenvalue continuous and/or differentiable?

I want to understand why the following function, the largest eigenvalue of a symmetric linear operator, is continuous and Gâteaux differentiable. \begin{equation*} \lambda(V)=\sup_{f \in \ell^2(I):\ ...
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1answer
30 views

Let $A$ be a non-separable $C^*$-algebra. Is it possible that there is a faithful representation $\pi:A\to L(H)$ on a separable hilbert space $H$?

Let $A$ be a non-separable $C^*$-algebra. Is it possible that there is a faithful representation $\pi:A\to L(H)$ on a separable hilbert space $H$? I know that if $A$ is separable, one can choose $H$ ...
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1answer
33 views

Does an essentially self-adjoint operator have the same kernel as its closure?

Let $H$ be a Hilbert space and let $A : D(A) \subset H \to H$ be an essentially self-adjoint operator. Let $\overline A$ be the unique self-adjoint extension of $A$. Question: Is it true that ...
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1answer
29 views

definition of block diagonal operator on a hilbert space

I 'm stuck with the definition of block diagonal operators on hilbert spaces. Def.: A bounded linear operator $T$ on a hilbert space $H$ is called block diagonal if there exists an increasing ...
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0answers
25 views

Idempotent and positive definite operator implies self adjoint

Let $H$ be a Hilbert Space (over $\mathbb{R}$ or $\mathbb{C}$ but maybe is valid for any field) and $E$ a continuous operator. Suppose $E$ is idempotent, i.e.,$E^2=E$ and positive definite, i.e. ...
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2answers
43 views

What is the $C^*$-algebra generated by a normal operator?

The following is the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I don't find the definition for the $C^*$-algebra generated by a normal operator in the book. ...
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1answer
39 views

Weak Solutions to PDES

I am working through some practice problems for my PDE class and I came across the following: Let $U\in \mathbb{R}^n$ be a smooth, bounded, connected open set. Let $\Gamma_1$, $\Gamma_2$, be two ...
4
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3answers
43 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
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1answer
40 views

Projection Theorem

I've been trying to apply the projection theorem to the following problem with no success. I've spent a few hours on this today, any help would be appreciated. Let H be a finite dimensional Hibert ...