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

learn more… | top users | synonyms

1
vote
2answers
114 views

Every bounded sequence has a weakly convergent subsequence: salvage this proof?

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
1
vote
0answers
11 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
1
vote
1answer
21 views

Question about the following proof in Hilbert space

I started reading the book "Mixed Finite Element Technologies" by Peter Wriggers and Carsten Carstensen, and I have a question about the following. Here is the setup: Then, the authors prove the ...
1
vote
0answers
33 views

Is a cyclic subspace of a compact unitary representation finite dimensional?

Let $K$ be a compact Lie group and let $\rho_k: H \rightarrow H$ be a (strongly continuous) unitary representation of K on a Hilbert space H. Why does the orbit, $\rho(K)v$ ,of any $v\in H$ generate a ...
2
votes
1answer
32 views

Mourre Theory: Resolvent Formula

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its resolvent by: $$z\in\rho(H):\quad R(z):=(z-H)^{-1}$$ Introduce its ...
0
votes
2answers
35 views

Is the Norm of the Square Root of an Operator equal to the Square root of the Norm of the Operator

Suppose we have a positive operator $A \in \mathcal{B}(\mathcal{H})$, does it follow that $$\|A\|^{1/2} = \|A^{1/2}\|?$$ If not, is there some relation between these quantities?
1
vote
0answers
37 views

What are the negative-dimentional n-sphere and n-cube?

The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres. $$\begin{array}{ll} S_{n-1}(R) &= ...
1
vote
2answers
18 views

Why $\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$?

Let $f,g\in L^2$ with Lebesgue measure. and $K:L^2\to L^2$ be some linear and continuous operator. Show that $$\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$$ where $h\in H\subset L^2$.
1
vote
1answer
29 views

Question about bilinear form on Hilbert space

I am trying to verify the following Let $H$ be a Hilbert space, and let $a(\cdot,\cdot)$ be a real continuous bilinear form on $H$ Then, define the operator $A:H-> H'$ as $Au(v) :=a(u,v), v\in H$ ...
0
votes
2answers
49 views

Question about dual of dual of Hilbert space

Let $H$ be a real Hilbert space and let $H'$ be the set of continuous linear functionals on $H$. Then, I know by the Riesz Theorem that for every $L(\cdot) \in H'$, there exists a unique $u\in H$ so ...
0
votes
1answer
72 views

Existence and uniqueness of the minimizer of Moreau-Yosida approximation

Let $f:H\to\mathbb{R}$, where $H$ is a Hilbert space, be a function that is bounded below, convex ($f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) \text{ for all } x,y\in H \text{ and } 0\leq t\leq 1$), and lower ...
2
votes
1answer
52 views

Inner product in Hilbert spaces

Considering a sequence $\{\boldsymbol{v}_k\}_{k=1}^\infty$ in a Hilbert space $\mathcal{H}$, and let $\{c_k\}_{k=1}^\infty \in \ell^2(\mathbb{N})$. Then for all $\boldsymbol{v}\in\mathcal{H}$ $$ ...
0
votes
1answer
22 views

Adjoint operators in Hilbert space

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that $$ \langle XY \boldsymbol{v}, ...
2
votes
1answer
45 views

Functions so that image of min (resp. max) is a positive definite kernel

I am trying to determine the functions $\phi : \mathbb{R}^+ \to \mathbb{R}$ such that: Pb 1: $K(s, t) = \phi( \mathrm{min} (s,t))$ is a positive definite kernel on $\mathbb{R}^+$. Pb 2: $K(s, t) = ...
3
votes
0answers
41 views

If $u_n \rightharpoonup u$ in $L^2(0,T;L^2)$ and $u_n$ bounded in $L^\infty(0,T;L^2)$, does $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ a.e. $t$?

Let $u_n$ converge weakly to $u$ in $L^2(0,T;L^2(\Omega))$ and let $u_n$ be bounded in $L^\infty(0,T;L^2(\Omega))$. Is it true that $u_n(t) \rightharpoonup u(t)$ in $L^2(\Omega)$ (weakly) for a.a. ...
2
votes
0answers
28 views

M bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M \}<\infty$

Let $H$ be a Hilbert space. Show that $M\subset H$ is bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M\}<\infty$ for every $x\in H$ My attempt: Since $H$ is a Hilbert space any set is ...
1
vote
1answer
51 views

Boundness of funcions in $L^2(0,T;H)$

Let $H$ be a Hilbert space and $u_{k} \rightharpoonup u$ in $L^2(0,T;H)$ (the $\rightharpoonup$ means "weakly convergent to") Assume one has the uniform bounds $$\mathrm{essential~sup}_{0\leq t\leq ...
-1
votes
1answer
69 views

Norm of the product of an isometry and a bounded operator

Let $A$ be a bounded operator and $V$ a linear isometry, both defined on a complex Hilbert space $H$ (infinite dimensional). I could easily prove that $\|VA\|=\|A\|$. But, I just couldn't prove that ...
0
votes
1answer
29 views

computation with polar decomposition of bounded operator on hilbert space

I am trying to prove the following homework problem: Let $T \in B(H)$ (so $T$ is a bounded operator on a Hilbert space $H$), and let $T = U|T|$ be the polar decomposition of $T$. Prove that if $T$ is ...
0
votes
0answers
27 views

Properties of functional calculus

Suppose we have a self-adjoint bounded operator $S$ on a Hilbert space $\mathscr{H}$ with the property that $||Sx||<||x||$ for each $x\in\mathscr{H}\setminus\{0\}$. Now assume that ...
-1
votes
1answer
51 views

Semigroups: Product Rule [closed]

Given a Banach space $E$. Consider C0-semigroups: $$S,T:\mathbb{R}_+\to\mathcal{B}(E)$$ Then the product rule holds: $$(TS)'(t)x=T'(t)S(t)x+T(t)S'(t)x$$ How to prove this from scratch?
0
votes
0answers
28 views

Help with inverse mapping theorem in Hilbert Spaces?

The question I'm trying to answer goes as follows: Let $X$ and $Y$ be banach spaces and $T:X \to Y$ be a bounded linear operator which is surjective. Let $K$ be the closed subspace $\ker T$ and let ...
0
votes
1answer
17 views

How to show that if $u$ is a partial isometry then $u = u u^\ast u$?

Let $H, H'$ be Hilbert spaces, $u \in B(H,H')$ and $u^\ast $ its adjoint. I am trying to show that if $u$ is a partial isometry then $u = uu^\ast u$. My idea was to write $H = \ker u \oplus (\ker ...
1
vote
1answer
19 views

Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast\|$?

Let $H$ be a Hilbert space and $u \in B(H)$ and let $u^\ast$ denote its adjoint. I know that $\|u\|=\|u^\ast\|$. But now I am wondering: Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast h\|$ for ...
0
votes
0answers
29 views

spectrum, hilbert space, symmetric operator

I am having trouble with the following... Let $A$ be a symmetric operator on a Hilbert space which is not self adjoint. Show that $\sigma(A)=\mathbb{C}$ or $H^+=\{\mu+iv \mid v\geq 0\}$ or ...
0
votes
0answers
14 views

Scalar product inequality

I would like to prove the following inequality: $$\langle f,1 \rangle \sum_{i=1}^N \sum_{j=1}^i \langle f,\mathbb{1}_{I(j)} \rangle \leq \frac{N}{2} \langle f,1 \rangle^2$$ where $f$ is a step ...
0
votes
0answers
27 views

Modulus: Invariant Domain

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense subspaces: ...
1
vote
1answer
30 views

Show $\mathcal{H}_\eta = L^2([a,b], \eta)$ is a Hilbert space when $\eta$ is positive, not necessarily continuous

Exercise $8$ of Stein and Sharkarchi's Real Analysis asks first to show that the space of measurable $f$ on $[a,b]$ such that $$\int_a^b |f(t)|^2 \eta (t)dt < \infty $$ denotes $\mathcal{H}_\eta = ...
3
votes
1answer
34 views

$\langle f, \phi_n \rangle = 0 \implies f = 0$ is equivalent to the definition of orthonormal basis

Is there an "easy" way to see that if $\{\phi_n\}_{n=1}^\infty$ is a set of orthonormal functions in a Hilbert space then showing $\langle f, \phi_n \rangle = 0$ for all $n$ implies $f = 0$ is ...
3
votes
0answers
73 views

Spectral Measures: Helffer-Sjöstrand

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard almost analytic extensions: $$f_E\in\mathcal{C}^\infty_0(\mathbb{C}):\quad ...
0
votes
0answers
29 views

Examples of non-self-dual Hilbert spaces?

I'm looking for some basic examples of non-self-dual Hilbert spaces, as well as basic examples of self-dual complex Hilbert spaces. Concrete examples would be helpful.
1
vote
1answer
48 views

Spectral Measures: Stone's Formula

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Then Stone's theorem says: ...
2
votes
1answer
62 views

Orthonormal basis from Riesz basis

This question is with respect to Theorem 7.1 of Mallat's Wavelet Tour text. It is a follow-up of sorts to a previous question. Preliminaries Suppose I have a set $\{\theta(t-n)\}_{n \in \mathbb{Z}}$ ...
2
votes
2answers
57 views

Norm, adjoint operator and compactness os some operators

Let $A_i:\ell^2\rightarrow \ell^2$ be two operators given as follows: $A_1x=(0,x_1,0,\frac{x_2}{2},0,\frac{x_3}{3},...)$ and $A_2x=(x_1,x_1,x_2,x_2,x_3,x_3,...)$ Compute the norm and the adjoint ...
1
vote
1answer
37 views

Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$T:\mathcal{D}(T)\to\mathcal{K}:\quad ...
1
vote
1answer
61 views

Relatively compact sequence in $L^2$

Let $\{f_n\}_n$ be a sequence in $L^2(\mathbb R)$. Suppose that there exists a sequence of closed balls $B_k \subset \mathbb R$ such that, for all $n$, $$ \int_{\mathbb R - B_k} |f_n|^2 \leq ...
0
votes
1answer
25 views

little question; nonseperable Hilbert spaces: what kind of basis…?

It is well known that every separable Hilbert space has a countable orthonormal basis. This type of basis is a schauder basis. If the Hilbert space is nonseperable, the Hilbert space has a orthonormal ...
1
vote
1answer
41 views

Showing $u^\ast$ is selfadjoint: stuck

Let $H$ be a Hilbert space and $u \in B(H)$. Define $(h,h') \mapsto \langle u(h), h'\rangle$. I am trying to show that $u = u^\ast$ if $\langle u(x),y\rangle = \overline{\langle y,u(x)\rangle}$ but ...
0
votes
1answer
61 views

orthonormal basis for $L^2(\mathbb{R})$.

So if we consider $L^2(\mathbb{R})$ as an Hilbert space with inner product $(\cdot ,\cdot)$. Define $\psi_n(x)=e^{-\frac{x^2}{2}}H_n(x)$ where $H_n(x)$ is the Hermite Polynomials. Then how do you show ...
0
votes
1answer
35 views

Change of inner product on Hilbert space

Let $(\mathcal{V},\langle\cdot,\cdot\rangle_1)$ be a Hilbert space. If we change the inner product, can we then say anything about if that is a Hilbert space as well, i.e. when is ...
0
votes
1answer
44 views

How to understand “completeness” intuitively?

In my text, it says, "if cauchy sequence in a normed vector space converge, i.e. $$\lim_{j,k \to\infty} ||u_j - u_k|| = 0$$ then the normed vector space is complete". The definition of completeness ...
0
votes
1answer
27 views

Do I have to show this map is well-defined?

Let $H$ be a Hilbert space and $u \in B(H)$. Write $$ H = \overline{\mathrm{im}(u)} \oplus \overline{\mathrm{im}(u)}^\bot$$ and define $v(h) = v(|u|x \oplus z):= u(x)$. Do I have to prove that ...
0
votes
0answers
24 views

Discrete Laplace: ONB

Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ Regard the unitary transformation: ...
3
votes
0answers
53 views

Weak vs strong convergence for unitary operators

Suppose $H$ is a separable complex Hilbert space with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$, where $\|u\|^2 = (u,u)$. Suppose $u, u_1, u_2, \dots \in H$. Then $\lim_{n \to \infty} u_n = ...
1
vote
1answer
105 views

Why is image $u^\ast$ closed

Let $H,H'$ be Hilbert spaces and $u\in B(H,H')$. Let $u$ be Fredholm. Then there exists a pseudo inverse $v\in B(H',H)$ such that $u = u v u$ and $v$ is Fredholm. Then $u^\ast = u^\ast v^\ast u^\ast$ ...
-1
votes
2answers
51 views

Selfadjoint Operators: Characterization

Given a Hilbert space. Symmetric operators can be described by $$\overline{\mathcal{D}(A)}=\mathcal{H}:\quad A\subseteq A^*\iff\langle ...
2
votes
2answers
38 views

Bounded Linear Operator and the Adjoint

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset ...
2
votes
0answers
53 views

Show that the space $C^{0, \gamma}(U)$ is complete

How can we show that the space $C^{0, \gamma}(U)$ is complete?? I have tried the following: So that the space is complete, the following has to stand: $$\forall \epsilon >0, \exists n_0 ...
0
votes
0answers
90 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
0
votes
0answers
23 views

Inner product and bases of Hilbert Spaces

This question might end up being flagged as too general, I'm not sure. Consider the equality $\langle a_n|\psi\rangle=c_n$ where $|\psi\rangle = \sum_n c_n|a_n\rangle$ is a vector of a Hilbert space ...