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

learn more… | top users | synonyms

-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
27 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
28 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
69 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
28 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
44 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
61 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
56 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
51 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
97 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
50 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
37 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 ...
1
vote
0answers
18 views

Relationship between spectrum of $-\Delta:H^1(M) \to H^{-1}(M)$ and $-\Delta:L^2(0,T;H^1(M)) \to L^2(0,T;H^{-1}(M))$?

Let us take a compact Riemannian manifold $M$. Let us define $-\Delta:H^1(M) \to H^{-1}(M)$ by $$\langle -\Delta u, v \rangle = \int_M \nabla u \nabla v$$ and $-\tilde \Delta:L^2(0,T;H^1(M)) \to ...
0
votes
0answers
21 views

What geometric information can be recovered from $L^2(X)$ for a manifold $X$?

It is well known that a compact Hausdorff topological space can be fully reconstructed from its $C^\ast$-algebra of complex valued continuous functions with the sup norm. Are there similar (partial) ...
1
vote
1answer
41 views

How is the following expresson be obtained and the meaning of the expression in blue box?

Let me introduce the term {$E_\lambda:\lambda \geq0$} is the spectral resolution of identity of a self adjoint densely defined, positive and closed operator $A:D(A)\subset X\rightarrow X$ , Where X ...
0
votes
1answer
38 views

Question about the Image of a compact transformation of a Hilbert space

$T$ is a compact operator on a Hilbert space. Show that $\operatorname{im}(T)$ does not contain a closed infinite dimensional subspace. Here is my attempt at the problem: Suppose that ...
0
votes
1answer
18 views

Direct sum of unitary operators is unitary

Is it true that if I have a Hilbert space $X$ that can be written $X = A \oplus B$ and unitary operators $T_1: A \to A$, $T_2: B \to B$ then the operator $T: A \oplus B \to A \oplus B$ given by ...
0
votes
1answer
19 views

Follow up on Hilbert spaces: chicken egg problem with projection and is the kernel always closed?

I am having a chicken egg problem with projections in Hilbert spaces. I was trying to show that if a Hilbert space can be written as $H = U \oplus U^\bot$ where $U$ is any subspace then $U$ is closed. ...
1
vote
1answer
40 views

On writing Hilbert spaces as sum of orthogonal complement

Is there a theorem that says: If $H$ is a Hilbert space and $U$ is any subspace then $$ H = U \oplus U^\bot$$ if and only if $U$ is closed? My conjecture is yes. I can easily prove that if ...
0
votes
2answers
17 views

Orthogonal and algebraic compement inclusion

Let $H$ be a Hilbert space. Let $U$ be a subspace and let $E$ be any complement such that $$ H = U \oplus E$$ I am wondering if it can be said that $U^\bot$ is contained in $E$. If not can it ...
1
vote
1answer
30 views

On Cauchy sequences, infimum and the proof of Hilbert space projection theorem

Let $H$ be a Hilbert space and $C$ a convex subset. The Hilbert space projection theorem states that there exists a unique $c_0\in C$ with $\|c_0\| = \inf_{c \in C}\|c\|$. I'm confused about the ...
0
votes
1answer
25 views

Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes ...
1
vote
1answer
28 views

Tensor Product: Denseness

This is thread is just a note. Given Hilbert spaces. Then denseness will be inherited on tensor products: $$\mathcal{D},\mathcal{E}\text{ dense}\implies\mathcal{D}\otimes\mathcal{E}\text{ dense}$$ ...
1
vote
1answer
25 views

Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
0
votes
1answer
31 views

Tensor Product: Identification

This is meant as note. Given a measure space and a Hilbert space. Then there's an identification: $$\mathcal{L}^2(\mu)\hat{\otimes}\mathcal{H}\cong\mathcal{L}^2_\mathcal{H}(\mu):\quad ...
0
votes
1answer
27 views

Tensor Product: ONB

This thread is just a note. Given Hilbert spaces. Consider their hilbertian tensor product: ...
4
votes
2answers
32 views

If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
1
vote
1answer
47 views

Showing $(M^\perp)^\perp=\overline{M}$

I have a question about a step in proving $(M^\perp)^\perp=\overline{M}$ where $M$ is a linear subspace of a normed vector space $E$. And $M^\perp=\{f\in E^*|\langle f,x\rangle =0\}$ This is the ...
0
votes
0answers
39 views

Invertible operators on a separable Hilbert space

Using polar decomposition or Kuiper's theorem one can show that the set of invertible operators on a separable Hilbert space $H$ is a connected subset of ${\mathcal B}(H)$. But does anyone know an ...
2
votes
1answer
35 views

Bergman space $L_a^2(\mathbb C)$

I claim that the Bergman space $L_a^2(\mathbb C)$ is the zero space. Is this true? If it is, how can I prove that every non-constant entire function is not in $L^2$?
1
vote
0answers
30 views

Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
4
votes
1answer
62 views

Fourier multiplier is the only translation invariant bounded linear operator on $L^2[-\pi, \pi]$

This a question from Stein-Shakarchi Real Analysis. Let $\mathcal{H}= L^2[ -\pi, \pi]$. And define the $\textbf{Fourier Multiplier}$ by, $$Tf(x) \sim \sum_{-\infty}^{\infty} \lambda_n a_n e^{inx}$$ ...