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

learn more… | top users | synonyms

1
vote
0answers
19 views

Volterra-like operator is bounded

Define $T:L^2(\mathbb R) \rightarrow L^2(\mathbb R)$ by $$(Tf)(x)=\int_{-\infty}^x e^{-(x-y)} f(y) \, dy.$$ I would like to show that $T$ is bounded and that $$\lambda = \frac{1}{1+iw}$$ is in its ...
0
votes
1answer
33 views

A closed subspace of a separable Hilbert Space is Separable

Suppose $X$ is a Hilbert Space which is separable. Let $Y$ be a closed Subspace of $X$. I need to show that $Y$ is separable. Since $X$ is separable it has a countable dense subset say $M$. Taking ...
2
votes
0answers
20 views

Orthogonal Complement: Families

Problem Given a Hilbert space $\mathcal{H}$. Consider a family: $$A:\Lambda\to\mathcal{P}(\mathcal{H}):\lambda\mapsto A_\lambda$$ Remind that: $$A\subseteq\mathcal{H}:\quad ...
0
votes
0answers
17 views

Calculate the real and the imaginary part of $g_n=\phi_n-\sum_{i=0}^{n-1}\langle\phi_n,\phi_i\rangle g_i$

We have $\{\phi_n\}_{n=0}^\infty$ a linearly dense sequence of unit vectors in a Hilbert space $H$ (on $\mathbb C$). Define $$g_n=\phi_n-\sum_{i=0}^{n-1}\langle\phi_n,\phi_i\rangle g_i$$ Calculate ...
1
vote
1answer
21 views

To show that $y$ is the best approximation of $x$ from $G$ i.e $y$ is the unique element of $G$ such that $||x-y||=d(x,G)$

Let $G$ be a closed subspace of a Hilbert Space $H$. For $x \in H$, let $y$ be the orthogonal projection of $x$ on $G$. Then I need to show that $y$ is the best approximation of $x$ from $G$ i.e $y$ ...
1
vote
1answer
46 views

$(T_n)_{n\in\mathbb{N}}\subseteq L(H)$, $T_n\to T$ weak, why does there exist $C>0$ such that $\|T_n\|\le C$ for all $n\in\mathbb{N}$?

Let $H$ be a Hilbert space, $(T_n)_{n\in\mathbb{N}}\subseteq L(H)$ a sequence such that $T_n^*=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a map $T\in L(H)$ such that $T^*=T$ ...
1
vote
1answer
16 views

Invariant subspace and projection

Let $F$ be a subspace of a Hilber space $H$, invariant under a bounded linear map $T$, and let $P$ be an orthogonal projection such that $Im(P)=F$. I need to show that $F$ and $F^\perp$ are ...
2
votes
1answer
33 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
0
votes
1answer
25 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
8
votes
1answer
107 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
0
votes
0answers
9 views

Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
-1
votes
0answers
22 views

A is an Hilbert operator, $(A+I)^{-1}$ is continuous with dense domain then A is essentially self-adjoint

We have no information about A, just that is an operator defined in $X\subset H$ where $H$ is a Hilbert space.
3
votes
1answer
48 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
0
votes
2answers
23 views

Complete eigen-vector basis from non invertible linear application

Consider a non-invertible linear application $O$ acting on a Hilbert space (quantum mechanics). Is there still any chance to find a complete basis of $O$ eigen-vectors or no?
0
votes
1answer
19 views

Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...
5
votes
2answers
75 views

Does orthogonal decomposition characterize direct sums in Hilbert space?

Let $H$ be a Hilbert space with inner product $(\cdot, \cdot)$. I know that if $M$ is a closed subspace of $H$, then $H$ can be written as the direct sum $M \oplus M^\perp$, where $M^\perp$ stands ...
2
votes
1answer
48 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
0
votes
0answers
27 views

Dual of Hilbert space : induced norm vs. operator norm

Let $\mathfrak{H}$ be a Hilbert space. Is the operator norm on the dual $\mathfrak{H}^*$ induced by a inner product ?
0
votes
1answer
17 views

$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality $$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle ...
2
votes
1answer
28 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
0
votes
0answers
17 views

Simple inner product relation for self-adjoint operators

Problem 6.3.7 in Friedmann's Foundations of Modern Analysis asks to show that if $A$ is a self-adjoint operator in a Hilbert space then \begin{align} 4(Ax,y) = &\left[\left(A(x+y),x+y\right) - ...
1
vote
1answer
41 views

Need help proving $n(T)=n(T^*)$ for finite dimensions.

In my book this is showed: Let H and K be complex Hilbert spaces and let $T\in B(H,K)$. There exists a unique operator $T^* \in B(K,H)$ such that $(Tx,y)=(x,T^*y)$ for all $x\in H$ ...
2
votes
1answer
52 views

A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
3
votes
1answer
59 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
3
votes
2answers
190 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
2
votes
0answers
28 views

Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
1
vote
1answer
17 views

Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does ...
1
vote
3answers
75 views

Stone's theorem for bounded operators

Let $H$ be a Hilbert space (assume separable if you like), and let $(U_t)_{t\in\mathbb{R}}$ be a unitary representation of $\mathbb{R}$ on $H$. Let us assume that $t\mapsto U_t$ is continuous, where ...
2
votes
2answers
27 views

Adjoint of $\lambda I - T$

Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$. I am tempted to argue as follows: ...
3
votes
1answer
69 views

Spectral resolution of multiplication operator

Kosaku YOSIDA claims in his book "Functional Analysis" that it is easy to see that the multiplication operator $Hx(t) = tx(t)$ in $L^2(-\infty,+\infty)$ admits the spectral resolution $H = ...
2
votes
0answers
28 views

Operator norm of $P[v]-P[w]$

Let $\mathcal H$ be a complex Hilbert space with inner product $\langle\mid\rangle$, (dirac notation) which is semi-linear (conjugate linear) in the first argument. Let $\mathcal P_1=\mathcal P_1 ...
3
votes
1answer
30 views

Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
0
votes
1answer
43 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
2
votes
1answer
46 views

Show that T is compact if $T^*T$ is compact

Let $H$ be a Hilbert space and $T: H\to H$ be a bounded linear operator. $T^*$ is the Hilbert adjoint operator. Show that $T$ is compact if $T^*T$ is compact. I am stuck with this proof. Any help ...
4
votes
0answers
43 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
2
votes
0answers
16 views

Verification of a theorem regarding Mercer Kernel.

if $\langle x,y\rangle$ is a Mercer kernel, then is $\langle c_1 x_1 + c_2 x_2,y\rangle$ a Mercer kernel where $c_1+c_2=1$? Ans: I give the following (dirty) line of proof. Please tell whether its OK ...
0
votes
0answers
31 views

Uncountable Kronecker Delta?

If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice). Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set ...
2
votes
2answers
25 views

not understanding a step in a proof

Hi: I'm reading some introductory notes on hilbert spaces and there is a step in a proof that I don't follow. I will put the exact statement below. If someone could explain how it is obtained, it's ...
0
votes
0answers
27 views

Normal Operators: Superalgebra (II)

Problem highlighted at the end! Application Reduction to only one operator!! Reference This builds up on: Superalgebra (I) Convention All operators possibly unbounded!! Structures Given a ...
6
votes
2answers
84 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
2
votes
2answers
47 views

Second differential of the norm in an infinite dimensional Hilbert space

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product [solved] So we can ...
1
vote
0answers
24 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
1
vote
1answer
30 views

Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
1
vote
0answers
21 views

Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
0
votes
1answer
29 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
4
votes
1answer
96 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
0
votes
1answer
32 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha:\mathcal{D}H_\alpha\subseteq\mathcal{H}_\alpha\to\mathcal{H}_\alpha:\quad H_\alpha=H_\alpha^*$$ And their ...
3
votes
3answers
60 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
7
votes
2answers
110 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
0
votes
1answer
69 views

Trace: Independence

Problem Given a Hilbert space $\mathcal{H}$. Consider an operator: $$A\in\mathcal{B}(\mathcal{H}):\quad\operatorname{Tr}|A|<\infty$$ Regard ONB's: ...