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

learn more… | top users | synonyms

4
votes
2answers
168 views

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

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
4
votes
1answer
59 views

Selfadjoint and continuous operator on a complex Hilbert space

Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show: $$ \lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H. $$ -- How can I ...
4
votes
1answer
634 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
4
votes
1answer
816 views

How to prove this integral operator is compact?

$T_kf=\int K(x,y)f(y)dy$ where $K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}}$ $\phi(x)$ is a smooth function on a compact support. $f$ is defined on $R^n$ and $K$ is defined on $R^n\times R^n$ ...
4
votes
2answers
161 views

Orthogonality checking in Kreyszig exercise

Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
4
votes
1answer
83 views

Invertible operator not preserving Hilbert dimension

It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide. I am asking for an example of an invertible (bounded) linear operator ...
4
votes
1answer
389 views

Dense subset of a Hilbert space

Let $A$ be a dense subspace of a Hilbert space $H$. Denote $\ell^2$ the Hilbert space of (complex valued) square-summable sequences and denote $\ell^2(H)$ the Hilbert space of $H$-valued ...
4
votes
1answer
133 views

The union of cyclic subspace is also a cyclic space

Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any ...
4
votes
1answer
252 views

Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?

I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
4
votes
3answers
411 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
4
votes
1answer
80 views

For a Hilbert space $\mathcal{H}$, is every bounded linear operator on $\mathcal{H}$ a linear combination of unitary operators?

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space, and let $B \in \mathcal{B}(H)$ be a bounded linear operator on $H$. If $\mathcal{H}$ is a complex Hilbert space, then $B$ can be written as a ...
4
votes
1answer
159 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
4
votes
1answer
120 views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
4
votes
2answers
40 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
4
votes
1answer
100 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
4
votes
2answers
67 views

Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
4
votes
1answer
44 views

Direct product of a family of Hilbert spaces

Let $\{H_i\}_{i\in I}$ be a family of Hilbert spaces, defined $$H = \{f\in \Pi_{i\in I} H_i, \sum_{i\in I}|f(i)|^2<\infty\}$$ and inner product $\langle f,g\rangle : = \sum_{i\in I} \langle ...
4
votes
2answers
264 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
4
votes
3answers
287 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have ...
4
votes
1answer
136 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
4
votes
1answer
57 views

Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
4
votes
1answer
70 views

Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
4
votes
1answer
150 views

Multiplication operator and trace class

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators. ...
4
votes
1answer
363 views

Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthnormal families, $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H then they have the same carnality. So If I ...
4
votes
1answer
60 views

Help with proof of closedness of a set

Let $u_n$ be a sequence in Hilbert space such that $\|u_n\|=1$ for all $n$, and $\langle u_n|u_m\rangle=0$ whenever $n\neq m$. Why is the following set closed: $\{0\}\cup \{u_n \mid n\geq 1\}$? Thanks ...
4
votes
1answer
124 views

Showing a representation is irreducible by showing that a degenerate subspace has codimension one.

Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle. I'm trying to understand this implication. Basically we want to show that a certain representation ...
4
votes
1answer
422 views

Weak convergence

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
4
votes
1answer
143 views

An interesting condition for the completeness of an orthonormal system in $ L^2([0,1]) $

Let $\{u_n\}$ be an orthonormal system in $L^2([0,1])$, prove that $\{u_n\}$ is complete iff $$ \sum_{n=1}^\infty \intop_0^1 \left|\intop_0^x u_n(t)\;dt\right|^2 dx = 1/2.$$ It should be noted that ...
4
votes
3answers
3k views

How to prove that square-summable sequences form a Hilbert space?

Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space ...
4
votes
1answer
90 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
4
votes
1answer
85 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 ...
4
votes
1answer
39 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
4
votes
1answer
43 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
4
votes
1answer
254 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
4
votes
1answer
182 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
4
votes
1answer
240 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
4
votes
1answer
231 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
4
votes
2answers
376 views

Reproducing Kernel Hilbert Spaces for Dummies

I am in the middle of some machine learning paper that states that for function $f$, imposing the norm constraint, $\|f \|=1$, corresponds to an orthogonal projection onto the direction selected in ...
4
votes
1answer
449 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
4
votes
3answers
703 views

Question about example of non-separable Hilbert space

I have come across the following example of a non-separable Hilbert space: Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ ...
4
votes
1answer
124 views

Convergence of a series involving inner products

Let $\{A_{j}\}$ be a sequence of bounded operators on a Hilbert space satisfying $\|A_{j}^{\ast}A_{k}\| \leq C_{j - k}$ and $\|A_{j}A_{k}^{\ast}\| \leq C_{j - k}$ where $\sum C_{i} < \infty$. Fix an ...
4
votes
0answers
53 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
4
votes
1answer
119 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 ...
4
votes
0answers
105 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
4
votes
0answers
108 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
4
votes
0answers
88 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
4
votes
1answer
38 views

Is this a correct argument for why this space of sequences is a Hilbert space?

I am assigned the following problem (a piece of Exercise 6.5 in Brezis's book): Let $(\lambda_n)$ be a sequence of positive numbers such that $\lim_{n\to\infty}\lambda_n=+\infty$. Let $V$ be the ...
4
votes
1answer
72 views

In a separable Hilbert space, can you write an operator from $\mathcal H$ to $\mathcal H$ as a column-finite matrix?

In this question, we are representing an operator $T$ as a matrix with respect to an orthonormal basis $\left\{e_n : n \in \mathbb{N}\right\}$. To do so, we let $t_{ij} = \langle T(e_j),e_i\rangle$. ...
4
votes
0answers
100 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
4
votes
0answers
127 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...