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

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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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2answers
45 views

Dense Subspaces: Intersection

Hilbert Space: $\mathcal{H}$ Dense Subspaces: ...
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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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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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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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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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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 ...
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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: ...
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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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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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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
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 $H,$ ...
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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 ...
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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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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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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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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
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
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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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
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 ...
0
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0answers
9 views

Generalized Mixed Models and Hilbert Spaces

I recently came across the derivation of the normal equations for linear regression using Hilbert Spaces and projection theorem, and thought it was pretty cool. After doing a lot of googling, it seems ...
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2answers
59 views

Are those two definitions of orthogonal projection equivalent in a general Hilbert space?

I am taking a graduate level course in probability and we started off with some results in functional analysis. One thing that I feel I do not understand properly is the definition of an orthogonal ...
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0answers
14 views

$A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
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2answers
34 views

Complete orthonormal system in a finite dimension Hilbert space

I have to solve the following problem of functional analysis. Let $H$ be a Hilbert space of dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is ...
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64 views

Results about Hilbert-Sobolev space with homogeneous boundary condition.

I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in ...
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0answers
27 views

Exercise on Hilbert spaces and complete orthonormal systems

Let $H$ be a Hilbert space of finite dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is linearly isometric to $\mathbb{R}^N$. I can't start with this ...
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1answer
15 views

$T+i\operatorname{Id}$ is an isomorphism for self-adjoint $T$

Let $T:H\to H$ be a self-adjoint continuous operator on a complex Hilbert space. Claim: $T+i\operatorname{Id}$ is an isomorphism and $\|(T+i\operatorname{Id})^{-1}\|\leq 1$. A few observations: ...
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1answer
28 views

$\operatorname{span}\{x_n:n\in \Bbb N\}$is dense if $\sum_{n=1}^\infty \|x_n-e_n\|^2<1$

Let $H$ be a Hilbert space with orthonormal basis $\{e_1,e_2,\cdots\}$. Suppose $(x_n)$ is a sequence in $H$ with $\sum_{n=1}^\infty \|x_n-e_n\|^2<1$. Claim: The span of the $x_n$ is dense in ...
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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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1answer
51 views

Is there a relation between Cartesian and tensor product of function spaces and function factorizability

H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors ...
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2answers
23 views

If the sum of two weakly convergent sequences strongly converges, do the summands strongly converge?

Let $a_n \rightharpoonup a$ and $b_n \rightharpoonup b$ weakly in $H$, a Hilbert space, and suppose that $a_n + b_n \to a+b$ strongly in $H$. Is it true that $a_n \to a$ and $b_n \to b$ strongly? I ...
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2answers
47 views

What's the value of $\alpha$ satisfying $||f'||^2\ge \alpha||f||^2$? [duplicate]

I am reading a paper about numerical analysis of a certain method for solving operator equation. Let our Hilbert space be $L^2[0,1]$, we define the subspace $D\in L^2[0,1]$ by $$ D:=\{f\in ...
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1answer
30 views

Continuity of inner product and change of limit order

First of all, please note that the specific context of ergodic theory could possibly not matter and this could reduce to a simply a question about Hilbert Space. As a part of a proof i'm working on, ...
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33 views

Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
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1answer
255 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2(\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
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2answers
29 views

Exercise on separable Hilbert spaces and orthonormal system

Let $H$ be a separable Hilbert space and $\{e_n\}$ a complete orthonormal system of $H$. Prove that, if $\{y_k\}$ is a bounded sequence in $H$, the condition $\lim_{k\rightarrow\infty}(e_n, y_k)=0$ ...