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

learn more… | top users | synonyms

4
votes
1answer
232 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
0
votes
1answer
75 views

Two versions of Lax-Milgram theorem

I'm having some troubles differentiating between two versions of Lax-Milgram theorem, one shown in my class and one that I saw is common on the internet. Let $H$ be hilbert space, $B$ bilinear form ...
0
votes
2answers
51 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
1
vote
1answer
59 views

perturbation by orthogonal projection

Let $G$ be an operator with discrete spectrum on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$. My ...
1
vote
1answer
19 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
1
vote
1answer
110 views

Partial Isometries: Introduction

Attention This question has been modified drastically. It is done so the answer below is still correct. It is done so to allow more specialized threads. Problem How do I deal with partial ...
0
votes
0answers
33 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
0
votes
1answer
26 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
2
votes
1answer
87 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
1
vote
1answer
58 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
4
votes
2answers
209 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
1
vote
1answer
57 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
3
votes
1answer
120 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
1
vote
0answers
49 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
0
votes
1answer
135 views

Bounded Linear functional is the orthogonal projection onto its range.

Suppose we have $P:H\to H$, where $H$ is a hilbert space and $P$ is bounded and linear. Assume that it satisfies $P^2=P$ and $P^*=P$ where $P^*$ is the adjoint. Show that $||P||\leq 1$, that ...
0
votes
1answer
88 views

Wave Operators: Hamiltonian

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the wave ...
0
votes
1answer
56 views

Wave Operators: Preliminary [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the wave operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
2
votes
1answer
78 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
2
votes
1answer
73 views

Closed Operators: Spectrum

Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Suppose one has: $$T=\overline{T}=T^{**}$$ Then it may happen: $$\sigma(T)=\varnothing,\mathbb{C}$$ What ...
1
vote
2answers
117 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
2
votes
1answer
46 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
1
vote
1answer
74 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
1
vote
1answer
53 views

Hilbert transform on $L^p(\mathbb{T})$

Let $\infty >p\geq 2$, then for $f\in L^p(\mathbb{T})$ (here $\mathbb{T}=[0,1)$), show that for any real-valued trigonometric polynomial $f$, we have $H(f^2-(Hf)^2)=2fHf$. The hint is to use the ...
1
vote
1answer
142 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
0
votes
2answers
71 views

Spectral Measures: Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
0
votes
1answer
71 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
0
votes
1answer
58 views

A sum of a closed subspace and a closed one-dimensional space is closed

I'm losing my mind over this question. For $H$ a Hilbert space, $A,B$ closed subspaces, and $B$ is of dimension $1$, I want to prove that $A+B$ is also closed. I'm looking for a straightforward ...
1
vote
1answer
42 views

How can I calculate the projection of a Hilbert space into a closed subspace?

I was woundering if there is an easy way to calculate the projection of a Hilbert space into a closed subspace. Obviously one could write $P:H->C$ that is given by $P(x)=d$ s.a $d=inf||x-v||$ for ...
5
votes
1answer
233 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
0
votes
1answer
66 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
0
votes
1answer
60 views

Example of a subspace S of a Hilbert space such that S^(⊥⊥) does not equal S?

I try to find an example of a subspace S of a Hilbert space H such that S^(⊥⊥) does not equal S. I know that subspace cannot be closed as for closed subspaces S^(⊥⊥)=S holds true. Does there exist ...
2
votes
1answer
120 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
3
votes
1answer
53 views

Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
1
vote
1answer
53 views

Dense convex proper subset of the Hilbert space $l_2$: $\{x|\sum x_i=0\}$ [duplicate]

Let's consider the space $l_2$ (all sequences $x$ with $\sum x_i^2 < +\infty$) and its subset $Z = \{x|\sum x_i = 0\}$. I want to prove that the closure of $Z$ is $l_2$, but I can't. I tried to ...
3
votes
1answer
115 views

Definition of unitary operators

Let $\phi, \psi \in \mathcal{H}$ be some element from a hilbert space $\mathcal{H}$ and $U$ a linear operator $U: \mathcal{H} \rightarrow \mathcal{H}$. Does $$ \forall \phi: \| U \phi \|^2 = \| \phi ...
0
votes
1answer
87 views

Spectral Measures: Reducibility

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: ...
0
votes
1answer
33 views

If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of T

If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of $T$ Compute $T^∗$ in terms of $A$ and $B$.
2
votes
1answer
65 views

WOT convergence to SOT convergence

Let $H$ be a Hilbertspace and $T_n \in B(H)$ a sequence of operators with $T_{n+1} \geq T_{n}$. I want to to show that if there is a self-adjoint $T\in B(H)$ with $T_n \stackrel{WOT}{\rightarrow}T$ ...
1
vote
2answers
129 views

Hilbert spaces of holomorphic functions

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps ...
3
votes
1answer
816 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
0
votes
2answers
130 views

Stone's Theorem Integral: Basic Integral

Disclaimer This thread has been renewed: Stone's Theorem Integral: Advanced Integral Problem Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly continuous ...
0
votes
1answer
20 views

For continuous Linear functional show $L \notin (C[0,1] , || . ||_2)^* $

For $L: C[0,1] \to \mathbb{C}$ denote the linear functional by $L(f) = f(0)$. Show $L \notin (C[0,1] , || . ||_2)* $ If I were to show $L \in (C[0,1] , || . ||_2)^* $ I would need to show that $L$ is ...
0
votes
0answers
43 views

Uniqueness of $n$-th root in Hilbert space

Let $H%$ be a Hilbert space and $A \in \mathcal L(H)$, $A = A^*$, $A \geq 0$. Let $B = \sqrt[n]A$, where $n \geq 3$, i.e. $B \in \mathcal L(H)$, $B \geq 0$, $B^n = A$. How to show that such operator ...
1
vote
0answers
40 views

If $V\subset H\subset V^*$ is a Gelfand triple, which is the natural inner product on $V^*$?

is there any natural way to define a inner product on $V^*$? First we could consider Riesz isomorphism $\mathfrak{R}:V\rightarrow V^*$, and define $\langle F, G\rangle_{V^*}:=\langle ...
0
votes
1answer
53 views

Sequence of unitary l.i. vectors such tha the sequence converges weakly to a non-zero vector, but not strongly

Let $\mathcal H$ be an infinite dimensional Hilbert space and let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of unitary linearly independent vectors. I know, using Bessel's inequality, that if the ...
1
vote
2answers
87 views

Square root of a bounded operator in Hilbert space

Consider the power series expansion $$ \sqrt{1-z} = 1+\sum\limits_{k=1}^\infty c_k z^k, $$ converging absolutely in the ball $|z| \leq 1$. Let $H$ be a Hilbert space and $A \in \mathcal L(H)$ a ...
6
votes
2answers
999 views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...
3
votes
1answer
50 views

Matrix space, with $\langle A,B\rangle=\text{tr}(AB^*)$ isn't Hilbert space, how can i find a counter example?

Generally, I'm having quite troubles thinking about counter examples. So I would love if someone could guide me into finding the example for the following question by myself (and not just giving it ...
1
vote
1answer
43 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

$H$ is a Hilbert space, $M$ is a self-adjoint bounded linear operator on $H$ with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all $n$, both $S_n$ and $S_n - ...
1
vote
1answer
29 views

Verifying a bound on the norm of an operator in $l_2$.

The problem: Define $L: l_2 \rightarrow l_2$ by $L(x_1, x_2, ...) = (y_1, y_2, ...)$, where $y_n = (x_1 + x_2 + ... + x_n)/n^2$. Show that $||L|| \leq (\sum_{n=1}^\infty 1/n^2)^{1/2}$. My proof: ...