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

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29 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 ...
10
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1answer
619 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
0
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0answers
36 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 ...
3
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1answer
51 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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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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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
42 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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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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0answers
41 views

series of linear operators

Let $\mathcal{B}(\mathcal{H})$ be the Banach space of bounded linear operators on a complex, separable, infinite-dimensional Hilbert space $\mathcal{H}$. It is well known that ...
0
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1answer
44 views

Common solutions to quadratic equations associated to self-adjoint matrices

Let $\mathcal{H}$ be a complex Hilbert space of dimension $d<+\infty$, and let $\{|n\rangle\}$ with $n=0,\cdots,d$ be an orthonormal basis in $\mathcal{H}$. Let $\mathbf{A}$ be a self-adjoint ...
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1answer
71 views

Infinite sum of bounded linear operators on a Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, separable, complex Hilbert space, and let $\mathbf{a}$ and $\mathbf{b}$ be bounded linear operators on $\mathcal{H}$ such that ...
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0answers
12 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 ...
2
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1answer
30 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 $ ...
2
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1answer
831 views

Poincaré inequality in unbounded domain

Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
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2answers
45 views

Dense Subspaces: Intersection

Hilbert Space: $\mathcal{H}$ Dense Subspaces: ...
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0answers
31 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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0answers
31 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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0answers
22 views

A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
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0answers
18 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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0answers
18 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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0answers
14 views

Why we can write the spectral density$ e_{\lambda}$ in the following forms?

Let $e_{\lambda}$ be a the spectral density of i.e. $$ e_{\lambda} = \frac{dE_{\lambda}}{d\lambda} $$ associated to a the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex ...
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0answers
27 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 ...
2
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1answer
28 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: ...
2
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1answer
246 views

Spectral Measures: Stone's Formula

Hilbert Space: $\mathcal{H}$ Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Spectral Measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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1answer
59 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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0answers
36 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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0answers
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 ...
2
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1answer
30 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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0answers
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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0answers
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? ...
0
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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 ...
6
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2answers
66 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 ...
0
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1answer
21 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 $H,$ ...
2
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3answers
163 views

Normal Operators: Polar Decomposition (Rudin)

On page 332 theorem 12.35b) of Rudin functional analysis is show that if T is normal then it has a polar decomposition $T=UP$. Does he mean that $P=|T|$? He's a bit ambiguous as to how he defines ...
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0answers
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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0answers
17 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
31 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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0answers
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, ...
0
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1answer
32 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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0answers
44 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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0answers
9 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
35 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
24 views

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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2answers
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 ...
0
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1answer
28 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 ...
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1answer
13 views

Unitary space: prove that

How I can start this problem? $ X $ is unitary space. Prove that if $M_1, M_2 \subset X: $ $M_1\neq \emptyset ,M_2\neq \emptyset$ and $ M_1 \subset M_2 $ then $ M_2^\perp \subset M_1^\perp $ Thank ...
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2answers
24 views

Do powers of contraction on Hilbert space converging to zero imply convergence of its adjoint to zero also?

In my functional analysis class I was met with the following problem: We suppose that $ \mathbb{H} $ is a Hilbert space and that T is a contraction operator on H (meaning $ ||T|| \leq 1 $ in the ...