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

learn more… | top users | synonyms

0
votes
2answers
46 views

Spectral Measures: Spectrum vs. 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
56 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
44 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
22 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
190 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
42 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
0answers
45 views

Does the orthogonal projection theorem guarantees uniqueness of the projected space?

Given a Hilbert space $H$, and linear map $P:H \to H$ such that $P^2=P$ and for every $x\in H$ : $\|Px\| \le \|x\|$, there is a closed linear-subspace $M$ such that $P=P_M$, the projection on $M$. My ...
0
votes
1answer
45 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
64 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 ...
0
votes
0answers
46 views

If $A$ is a compact diagonal operator, with diagonal $\{\alpha_n\}$, then $\lim_{n\to\infty}\alpha_n=0$.

Here is my question: If $A\in \mathscr{B}(\mathscr{H})$ is a diagonal operator with diagonal $\{\alpha_n\}$, show that if $A$ is compact, then $\lim_{n\to\infty}\alpha_n=0$. Here is what I have: I ...
3
votes
1answer
40 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
40 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
45 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
26 views

Spectral Measures: Commuting Operators

The questions are given below!! Theorem Given a measure space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. Denote ...
0
votes
1answer
25 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
35 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
45 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 ...
1
vote
1answer
90 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
117 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
15 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
38 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
20 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
26 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 ...
0
votes
0answers
21 views

Matrices with Continuous Indices

The components of a matrix $A$ can be written as $a_{ij}$. In Quantum we're starting to talk about a generalization where the indices are not elements of $\Bbb N$, but are instead continuous. Our ...
1
vote
2answers
29 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 ...
1
vote
2answers
59 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
36 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
36 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
26 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: ...
0
votes
0answers
14 views

Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
1
vote
1answer
50 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
1
vote
2answers
59 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
2
votes
1answer
35 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
1
vote
1answer
41 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
5
votes
1answer
43 views

Showing a certain subspace of Hilbert space is dense

Let H be the Hilbert space of square-summable sequences of reals. A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they ...
3
votes
1answer
41 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
1
vote
1answer
34 views

Central Limit Theorem for transformed random variables

The Central limit theorem (CTL) is often given similar to the entry in Wikipedia as: Suppose ${X_1, X_2, ...}$ is a sequence of independent and identically distributed random variables with ...
1
vote
0answers
15 views

Coefficients in Representer theorem

I have a Mercer Kernel, $K\colon X \times X \rightarrow \mathbb{R}$, i.e. it is continuous, symmetric and postive definite on a compact domain $X \subset \mathbb{R}^n$. Also, I have a set of $m$ ...
8
votes
1answer
86 views

How are Hilbert Space methods used in number theory?

In N. Young's book An Introduction to Hilbert Space, there is an interlude in which the author remarks that the theory of Hilbert spaces is "routinely used in differential geometry, complex analysis, ...
1
vote
2answers
41 views

Basis of intersections of $L^p$ spaces

I keep confusing myself about a subspace basis and I can only find intelligible material discussing the finite, linear algebra, case. It is known that the Hilbert space $L^2(X)$ has a basis, for ...
2
votes
3answers
50 views

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$.

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$. I've been stuck on this for a while and don't really know where to start.
0
votes
1answer
26 views

Adjoint operator between Hilbert spaces, does it map a subspace onto a subspace?

Let $X$ and $Y$ be Hilbert spaces and let $f\colon X \to Y$ be a linear continuous bijection. Define the adjoint $f'\colon Y \to X$ by $(f'y, x)_X = (y, fx)_Y$. Let $X_0$ and $Y_0$ be subspaces of ...
2
votes
2answers
28 views

Completeness of the space of random variables with bounded conditional first moment with respect t0 $\left\Vert \cdot\right\Vert _{2} $

Consider a probability space $\left(\Omega,\mathcal{F},P\right) $, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F} $. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right) $ be the space ...
1
vote
0answers
25 views

Eigenvector of a linear combination of operators is an eigenvector of each operator

Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an ...
1
vote
1answer
31 views

Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
1
vote
0answers
54 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...
0
votes
1answer
12 views

Riesz map on $L^2(0,T;H)$ — it's not “unique” in a way

Let $R:L^2(0,T;H) \to L^2(0,T;H^*)$ denote the Riesz representation map. Given $u \in L^2(0,T;H)$, $Ru \in L^2(0,T;H^*)$ can be changed in $[0,T]$ on a set of measure zero. So $Ru$ is not unique in ...
3
votes
0answers
51 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
1
vote
1answer
39 views

Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
4
votes
1answer
93 views

T is not compact operator

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...