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

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Compact Operator with Infinite rank Doesn't have a Closed Image

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. Claim: A compact operator $T$ which has infinite-rank has an image that isn't closed. I'm trying to prove this claim but I'm ...
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103 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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2answers
33 views

Prove that $(A^\bot)^\bot = \overline{\operatorname{span}A}$

I proved that $(A^\bot)^\bot \supset \overline{\operatorname{span}A}$. To prove the reverse inclusion, let $x \in (A^\bot)^\bot$, then $\langle x, y\rangle = 0$ for all $y \in A^\bot$. How can I ...
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8 views

Parameterizing the set of subquotients of a Hilbert space

If you want to parameterize the set of subspaces of a finite-dimensional Hilbert space $V$, and naturally induce a norm derived from $V$ on the resulting moduli space, the classical approach is to ...
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2answers
35 views

Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
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1answer
20 views

spectral projection

Let $T$ be a self-adjoint operator on a Hilbert space $H$. $P$ is a projection on $H$. Let $E^{|PTP|}(1,\infty)$ be a spectral projection of $|PTP|$. My question is: whether $E^{|PTP|}(1,\infty) \le ...
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1answer
18 views

If $U,H$ are Hilbert spaces, $Q$ is an operator on $U$ and $U_0:=Q^{\frac 12}U$, find an expression for the norm of Hilbert-Schmidt operators $U_0→H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for ...
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1answer
28 views

If $Q$ is an operator on a Hilbert space $U$ and $(e_n)_{n\in\mathbb N}$ is an ONB of $U$ with $Qe_n=λ_ne_n$, then $Q^{-1}e_n=\frac 1{λ_n}e_n$

Let $U$ be a Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all ...
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1answer
37 views

completion of $C^{\infty}\left(S\right)$ is $L^2(S)$?

I have the space of infinitely derivable functions, i.e. $C^{\infty}\left(S\right)$ with the following inner product $$\left\langle f,g\right\rangle ...
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21 views

Tensor products in separable Hilbert spaces

In a variety of scientific publications, I have come across the use of tensor products of random Hilbert-Schmidt operators defined on separable Hilbert spaces. Let us introduce some notations. Define ...
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40 views

Self adjoint operators on Hilbert spaces are bounded

I think I have a proof that if $A: H\rightarrow H$ is a self adjoint operator on a Hilbert space $H$, then $A$ is bounded: We can use the closed graph theorem. Let $x_n \rightarrow x$ and $Ax_n ...
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30 views

Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by ...
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16 views

Positive definite functions coming from finite dimensional representations

Let $G$ be a topological group, let $\mathcal{H}$ be a complex Hilbert space, let $v\in\mathcal{H}$ be a nonzero vector, and let $\rho:G\rightarrow \mathcal{U}(\mathcal{H})$ be a unitary ...
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1answer
59 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
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23 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes ...
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13 views

Validity of: if $H$ is pre-Hilbert and $V$ is a f.d. vector subspace, then $f^*$ exists and is unique

We were recalling some basic stuff about Hilbert spaces in class, and the professor gave the remark: If $H$ is a Hilbert space and $V$ is a finite dimensional vector subspace of $H$, then for all ...
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1answer
34 views

Why is it true that the multiplication operator in a reproducing kernel Hilbert space is always continuous?

In my functional analysis I was met with this seemingly trivial theorem on RKHS If $ \mathbb{H} $ is a reproducing Kernel Hilbert Space and we have a multiplier $ \phi $ meaning it satisfies $ ...
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1answer
44 views

Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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33 views

orthogonal decomposition Hilbert Space

I see this decomposition in a paper, where $\oplus$ is a orthogonal direct sum, $E(B/A)$ is the orthogonal projection of B onto A, and $B^\perp$ is the orthogonal complemente of B in any hilbert space ...
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1answer
20 views

If a contraction and its adjoint converge to zero both does that mean the contraction satisfies $ ||Th|| < h $

I just met this in my functional analysis on contractions which got me stumped: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \leq ...
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33 views

Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

I was recently met with this in my functional analysis class on which I am stuck: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| ...
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49 views

Dense Subspaces: Intersection

Hilbert Space: $\mathcal{H}$ Dense Subspaces: ...
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20 views

Formula connecting the resolvent opeartor andthe spectral density?

I want to know if it is a formula connecting the resolvent opeartor $(\lambda - T)^{-1}$ for a selft-adjoint operator $T$ and its spectral density $e_{\lambda}$. Thank you in advance
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20 views

Formula connecting the resolvent and the heat kernels

Using the well known formula connecting the resolvent and the heat operators associated to a selft-adjoint opeartor $A$ \begin{align} (\zeta - A)^{-1} = \int_{0}^{\infty} e^{-\xi t} \, e^{t A} dt; ...
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1answer
52 views

About a relation between isometries

If we have $(T_i)_{i=1}^N$, operators on a Hilbert space, that are also isometries and satisfy the following relation: $$\sum_{i=1}^NT_iT_i^*=Id\quad (1)$$ How can you prove that they must also ...
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30 views

Metrizability of space of unitary operators on Hilbert space

Let $\mathbb{H}$ be a (complex) Hilbert space. Define $\mathbb{P}$ as a projective space over $\mathbb{H}$, i.e. $\mathbb{P}=\left(\mathbb{H}\setminus\{0\}\right)/\sim$, where $f\sim g$ iff there ...
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1answer
29 views

Direct Integral: Scalars

Given a Borel space $\Omega$. Regard the Hilbert Space: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}_+:\quad\mathcal{H}:=\mathcal{L}^2(\Omega;\mu)$$ Denote the Borel Projections: ...
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1answer
69 views

Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

I was asked this in functional analysis class: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || Th || < ||h|| $ for all $ h \in H $. We are asked if ...
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37 views

vector-valued function space definition except for measure zero

I am wondering what's the correct way to mathematically describe the following problem. Say you have an object that can be defined as an open set $\Omega \in \mathbb{R}^d$, where the dimension $ ...
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19 views

Does strict contraction lead to convergence to zero in norm?

In my functional analysis class I was asked this question which got me stuck: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || T || < 1 $ in the ...
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1answer
36 views

Showing that $\langle T(u),T(v)\rangle = \langle u, v \rangle$ implies $T$ is a linear isometry

Let $T$ belong to $\mathcal{L}(H)$ (i.e., the set of linear operators from $H \mapsto H$ where $H$ is a Hilbert space). I need to show that $T$ is an isometry iff $\langle T(u),T(v) \rangle = \langle ...
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29 views

Uniqueness of element in infinite dimensional Hilbert space

Suppose $H$ is an infinite Hilbert space where $\{e_k:k\in \mathbb{Z}\}$ is a total orthonormal family. Let $H_1=\overline{span{(e_k: k=0, 1,2,\cdots})}$ and $H_2=\overline{span{(e_{-k}+ke_k: ...
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22 views

Pondering on infinitely dimensional objects.

Suppose I have a Hilbert space M. Is it possible to define a set such that it resembles a geometric object in $\Re^n$?If so does this have a special attribute corresponding to a functional equation? ...
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1answer
18 views

Under what conditions are such operators well defined?

Let H be a hilbert space, and $\phi_k$ a basis, one can define a "diagonal" operator $A$ by $A\phi_k=b_k\phi_k$, Is there a simple condition on the coefficients $b_k$ such that the operator is well ...
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1answer
22 views

Equivalence of Hilbert spaces and application of dominated convergence theorem.

Let $H$ a separable Hilbert space with orthonormal basis $\{x_n\}$. Let $\{y_k\}$ a sequence of elements of $H$, show that the following statements are equivalents. (a) For all $x$ in $H,$ ...
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76 views

$P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
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14 views

From the direct sum theorem, how can we deduce that $y$ is the orthogonal projection of $x$?

In the direct sum theorem we have $$H =Y \oplus Z$$ where $Y$ is any closed subspace of a Hilbert space $H$. It is easy to deduce that for every $x \in H$ there is a $y \in Y$ such that $$x=y+z$$. ...
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16 views

In what sense are compact operators limits of finite-rank operators?

The convergence is in respect to what topology ? Can someone please write it mathematically ?
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20 views

Is there any connection between these two definitions of coercivity (ellipticity) in PDE and bilinear form?

In the field of partial differential equation, we say that the following operator \begin{equation} Lu=-\sum\limits_{i,j=1}^n (a^{ij}u_{x_i})_{x_j}+\sum\limits_{i=1}^n b^i u_{x_i}+cu \end{equation} is ...
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1answer
33 views

Strong convergence on the unit sphere of $l_2$

Let $(p_n)$ be a strictly increasing sequence of natural numbers and $(\epsilon_n)$ a positive sequence decreasing to $0$. Suppose $x_n$ is a sequence in $S(l_2)$ (the unit sphere of $l_2$) with the ...
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24 views

How to calculate the norm of this operator?

Let $H$ be a separable Hilbert space and $(\phi_k)$ be a basis $A(t)$ is defined such as $A\phi_k=\exp(-t/k)\phi_k$. I am specifically interrested whether $\|A(t)\| \to 0$ when $t \to \infty$ or not, ...
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10 views

If two compact, positive operators are close, are the projections onto subspaces also close?

Let $H$ be a Hilbert space. Let $a$ and $b$ be compact, positive operators acting on $H$. I wonder if the inequality $$\Vert \Pi_{\ker[a - \lambda_j(a)]}\, -\, \Pi_{\ker[b - \lambda_j(b)]}\Vert\leq ...
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2answers
37 views

$T$ is self-adjoint $\Rightarrow \exists$ positive $A,B$ such that $T=A-B$ and $AB=0$

I have a trouble by the following problem and I dont have any idea to solve it. can anybody give me a hint? Thanx in advance. Let $\mathcal H$ be a Hilbert space and $T:\mathcal H \to \mathcal ...
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1answer
76 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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1answer
38 views

Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
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47 views

Compact operators are orthogonally equivalent to a diagonal matrix?

On Brezis's Functional Analysis, the last question of Problem 44 (near the end of the book) reads (modified to include context) Assume that the Hilbert space $H$ is separable and $T\in\mathcal ...
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Hilbert spaces and quantum mechanics [duplicate]

how is Hilbert spaces applied in quantum mechanics? the differences between the application of C* -algebra and Hilbert spaces on quantum mechanics.
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37 views

Show that $S^{-1}: S(H) \rightarrow H$ exists

Let $$S=I+T^*T: H \rightarrow H$$ where T is linear and bounded. Show that $$S^{-1} : S(H) \rightarrow H$$ exists. I am working through Hilbert-Adjoint Operator exercises right now and am stuck with ...
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1answer
35 views

Spectral Measures: Poisson

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}H\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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1answer
42 views

Is the set $\ell ^2$ a $G_\delta$ in $\mathbb R ^\omega$?

$\ell ^2$ = set of sequences $(x_i)$ of real numbers such that $\|x\|=\sum _{i=0} ^\infty x_i ^2<\infty$. Question: Is $\ell ^2$ a $G_\delta$-set in the product topology $\mathbb R ^\omega$? I am ...