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Questions tagged [contraction-operator]

Use this tag for questions about operators whose norm is at most 1.

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Problems understanding integration in a banach fixed point exercise [closed]

I have been studying for a math test and I have encountered the following problem: banach's fixed point theorem applied to finding a unique function I do not get the last step. I do not know how ...
valentina manfredi's user avatar
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26 views

Can you use the interior product to calculate flux?

If I have a vector Field F and a 2D surface $\Sigma$ in $\mathbb{R}^3$ would $F \lrcorner dx \land dy \land dz= F_1 dy\land dz + F_2dz\land dx + F_3dx\land dy$ If this is the case, could one say that ...
Minimo's user avatar
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1 answer
49 views

Let $T$ be a contraction in a Hilbert space $H$. If $\| T^n x\| \to 0$ for $n \to \infty$, does $\| (c \cdot T)^n x \| \to 0$, where $c > 1$?

The obvious problem here is that because $c > 1$, then $c^n \to \infty$ for $n \to \infty$. So we have: $$\| (c \cdot T)^n x \| \le \underbrace{|c^n|}_{\to \infty}\cdot \underbrace{\|T^nx\|}_{\to 0}...
S-F's user avatar
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Is $C_0^{\infty}(\mathbb{R}^d)$ a core for diffusion operators on $C_0$?

This question is motivated by Hille-Yosida semigroup/Markov theory. Let $P_t:C_0(\mathbb{R}^d) \to C_0(\mathbb{R}^d), t \geq 0$ be a strongly continuous contraction semigroup. Say a function $f \in ...
Alex's user avatar
  • 517
2 votes
1 answer
133 views

$\sin(x)$ is a contraction

I see this question being answered multiple times on this website including here but it still seems inconsistent with the mean value theorem. If we have $\sin(x)$ defined on $[0,1]$ and we have $[x,y]\...
Stylel's user avatar
  • 91
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0 answers
46 views

Positivity of semigroup $(P_t)_{t\geq0}$ implies its contractivity, $\| P_tf\| \leq \|f\|$

Let $\Omega$ be a Polish space, denote $E = (C_b(\Omega),\|\cdot\|)$ be the Banach space of continuous and bounded functions with the usual supremum norm. A family $(P_t)_{t\geq 0} $ of linear ...
Jeffrey Jao's user avatar
2 votes
1 answer
150 views

proof that the operator of integral equation is a contraction

The part I am reading is similar to the following questions, so if you are not familiar with integral operator and contraction, you could refer to them. Show integral operator is contraction mapping ...
Halk's user avatar
  • 156
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1 answer
50 views

If $A \in \mathcal{B}(H), B \in \mathcal{B}(K)$ are contractions, is it then true, that $\begin{bmatrix}A&0\\0& B \end{bmatrix}$ is a contraction too?

$A \in \mathcal{B}(H)$ is a contraction, if: $$\| A x \| \le \| x \|$$ for every $x \in H$. In other words: $\| A \| \le 1$. The question here is: How can I show, that a matrix of operators (which ...
anon's user avatar
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What's an easy example of a family of three commuting contractions for which there's no isometric dilation but fulfill the von Neumann inequality?

Or the other way around, an example of a family of three commuting contractions for which there exists an isometric dilation, but which does not satisfy the von Neumann inequality? The only example I'...
anon's user avatar
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1 answer
94 views

Non-Symmetric Real Matrix with eigen values $\in $ [0,1] but largest singular value > 1

Matrix , M has all eigenvalues $\in$ [0,1], but on simulation, I can see that the largest singular value is >1. a) Can anyone give an example of such a matrix in toy cases like 2x2 b) Can some ...
Bhartendu Kumar's user avatar
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1 answer
35 views

Prove that the map $L(u,v) := (I+Q)^{-1}(u+v)-v$ is nonexpansive for skew-symmetric $Q$.

I am trying to understand the proof in the Appendix of the paper "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding" by O'Donoghue et al. (link: https://web....
Shourya Bose's user avatar
2 votes
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62 views

Solve $\ln(4x) = x$ using the Banach contraction theorem

I would like to find a solution of: \begin{equation} \ln(4x) = x \end{equation} My idea was to use the Banach contraction theorem: "If $E$ is a closed non empty set in a Banach space $V$ and $T: ...
user avatar
5 votes
1 answer
171 views

Conditions for $I-x A$ to be a convergent matrix for some $x\in \mathbb{R}$

I'm looking for interpretable necessary/sufficient conditions on $A$ which guarantee that $(I-x A)$ is a convergent matrix for some $x\in \mathbb{R}$ For instance, $A$ normal with non-zero eigenvalues ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
70 views

Find the metric so that this map is a contraction

Let $y_1, y_2 >0$ and $y_{n}=\frac{2}{y_{n-1}+y_{n-2}}$ for $n\ge 3$. I know that this sequence is converging and that $y_n \to 1$ as $n\to \infty$. Let us write $Y_n = (y_n,y_{n+1})$ and $f(a,b)=(...
J.Mayol's user avatar
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1 vote
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Skew-hermitian also a contraction?

If a matrix is skew-hermitian, is it also a contraction? I've been playing around with skew-hermitian operators on Hilbert spaces, and found this to be true on a couple of examples. So I was wondering ...
Andreas Burger's user avatar
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1 answer
51 views

Question regarding contraction theory

I am following the paper (A Study of Synchronization and Group Cooperation Using Partial Contraction Theory) to understand how contraction theory used to analyze coupled oscillators. On page 211, the ...
M.K's user avatar
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Fixed point iteration converges

I found an old problem from notes, which I was not able to solve. Assume that we have a given (arbitrary) norm $\| \cdot\|$ on $K$ and function $g:K \times K \rightarrow K \times K$ for some compact ...
ntrstd11's user avatar
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Unique fixed point of contraction defined on a ball

In the case where $f : X \rightarrow X$ is not a contraction on the whole space $X$, but rather a contraction on some neighborhood of a given point $y$, In this case we restrict our function to a ...
Alissia's user avatar
1 vote
1 answer
248 views

Trace and Lie derivative of a $(1,1)$-tensor commute (Direct proof)

My question is same as this MSE post but I want to use direct properties of Lie derivative and trace to prove (I know another proof using this fact that pullback map commutes with contraction) $$\...
C.F.G's user avatar
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1 answer
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Prove that second iterate of map is a contraction

Let $T: C[0,1] \rightarrow C[0,1]$ be a linear operator defined as $$ T f(x)= \int_{0}^x f(t)dt, \forall x \in [0,1]$$ Space $C[0,1]$ is equipped with supremum norm. Prove that $T^2$ is a contraction ...
User154's user avatar
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1 vote
0 answers
100 views

$\forall\ x,y\in [a,b], |f(x)-f(y)|<|x-y|$ implies that the limit exists...

Let $f: [a,b]\to [a,b]$ be continuous, and satisfy $\forall\ x\neq y\in [a,b], |f(x)-f(y)|<|x-y|$. For any $x_0\in [a,b], x_{n+1}=f(x_n)$, show that $\lim_{n\to\infty} x_n=\xi$, where $\xi$ is the ...
xldd's user avatar
  • 3,467
0 votes
1 answer
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Positive operators and Hilbert's metric

If an operator L on a finite-dimensional vector space is positive, then it is a contraction in the Hilbert metric $d(x,y)$. The contraction ratio is related to the projective diameter of the operator....
Augustin's user avatar
3 votes
2 answers
496 views

Contraction of a tensor over all indexes?

In Chapter 2.3 of this book, let $F \in T^k_l(V)$ and $\{\omega^i\}_{i=1}^l \in V^*, \{X_i\}_{i=1}^k$, then it is clear that $$ F \otimes \omega^1 \otimes \cdots \otimes w^l \otimes X_1 \otimes \cdots ...
macton's user avatar
  • 659
3 votes
1 answer
205 views

Contraction Mapping and Fixed Point with two different distance metrics

I have been looking at the fixed point theorems that use the contraction-mapping and all seem to use the same distance metric for the input and output spaces. If we have a differentiable mapping $f: X ...
GA-Student's user avatar
3 votes
3 answers
368 views

A question on contraction mapping theorem and fixed point iteration

First of all, thank you for taking the time to read my post. Secondly, this is a question I got as a part of homework. However, the professor allows us to work in groups so I'm hoping that this is ...
user avatar
0 votes
1 answer
25 views

Is the pullback of a contractive split again contractive?

If I have a pullback of $C^*$-algebras and I consider the underlying banach spaces Banach, in particular all maps in the pullback square are contractive, and I have a contractive split $s$ for one map,...
Roland's user avatar
  • 303
1 vote
0 answers
41 views

Contraction mapping theorem applied to integral equation

I am facing the problem of showing the existence of a solution to the following integral equation: \begin{equation}\label{Eq: IE} z(t) = e^{\lambda(T-t)-\beta\int_{t}^{T}z(s)^{\frac{1}{\beta-1}}ds}\...
Alessandro's user avatar
2 votes
0 answers
160 views

Difficulty showing that if $T:X\to X$ satisfies $\rho(T(u),T(v))\lt\rho(u,v)$ then $T$ must have a fixed point if $(X,\rho)$ is a compact metric space [duplicate]

Let $(X,\rho)$ be a non-empty compact metric space. I've learnt enough over the last week to know that that means it is complete, totally bounded and sequentially compact, and Cantor intersection ...
FShrike's user avatar
  • 41.2k
3 votes
1 answer
547 views

Sequence of contraction mapping and convergence of fixed point

Let $(𝑆,||_{\infty})$ be a metric space and $𝑇 : 𝑆→𝑆$ be a function mapping S into itself. $S$ is a space of bounded and Lipschitz continuous function. For each $𝑛\inβ„•$, $\tau_{n}\in T$ satisfies ...
Anonymously lost student's user avatar
2 votes
1 answer
259 views

Local Existence for Heat equation with Lipschitz nonlinearity

Let's say we have $x\in [0,1]$ and we have the heat equation $$\frac{d}{dt}u(x,t) = \Delta u(x,t) + f(u)$$ with $u(0,t) = u(1,t) = 0$ and $u(x,0)\in L^2$. $f$ is lipschitz continuous. I want to show ...
iYOA's user avatar
  • 1,408
0 votes
0 answers
131 views

When does contraction in sup norm implies pointwise contraction?

I was wondering about the following question : Suppose I have a contraction mapping $\tau$ for a bounded continuous function in supremum norm. $$ \beta \cdot ||u-v||_{\infty}\geq ||\tau(u)-\tau(v)||_{\...
Anonymously lost student's user avatar
2 votes
2 answers
304 views

Positive contraction operator on Hilbert space

This is an exercise from a functional analysis textbook. I am stuck in it. Let $H$ be a Hilbert space, $A\in\mathcal{B}(H)$. We call $A$ a contraction operator if $\|A\|\leq 1$ and call $A$ a positive ...
Stephen's user avatar
  • 786
2 votes
1 answer
195 views

Any result on ADMM iteration being a contraction?

Consider a standard ADMM problem: minimize $f(x) + g(z)$ subject to $A x + B z = c$ The scaled ADMM algorithm is (from Boyd's paper): $$ \begin{aligned} x^{k+1} &= \underset{x}{\operatorname{...
Truong's user avatar
  • 663
2 votes
1 answer
90 views

Two implications of an operator that preserves positivity on L2

Suppose $(X,\mathcal{A},m)$ is a probability space. Let A be an operator on function spaces. I am currently taking a course in measure theory, and in the lecture slides the following two remarks are ...
mathematics's user avatar
3 votes
0 answers
74 views

Question about integral operator being a contraction

I've been told that if $$T(f)(x)=g(x)+c\int_0^xK(x,y)f(y)dy$$ for some $c\in\mathbb{R}$ and $f,g\in C[0,1]$ ($g$ fixed) and $K\in C([0,1]\times[0,1])$, then there exists $n\in\mathbb{N}$ such that $T^...
123's user avatar
  • 25
1 vote
0 answers
98 views

Contraction mapping on functional space

I had some major modifications in my previous question. I first showed that $\tau$ satisfies Blackwell's conditions, which implies that $\tau$ is a contraction mapping. $\tau$ takes a bounded ...
Anonymously lost student's user avatar
1 vote
1 answer
51 views

What are the eigenvalues of $A(A+I)^{-1}$ in terms of the eigenvalues of SPSD matrix $A$?

Let $a_1, a_2 \dots$ be the eigenvalues of symmetric positive semidefinite (SPSD) matrix $A$. How to find the eigenvalues of matrix $A(A+I)^{-1}$?
Argha Chakraborty's user avatar
3 votes
0 answers
122 views

Show that Newton update in two variables converges

I am having two update rules of a two variable function $f(x, y)$, i.e., $$x \mapsto h_1(x,y):=x - \frac{f(x, y)}{\partial_x f(x,y)} \tag{1}$$ $$y \mapsto h_2(x,y):=y - \frac{f(x, y)}{\partial_y f(x,y)...
darkmoor's user avatar
  • 793
1 vote
2 answers
82 views

Convergence Behavior of Sequence with Diminishing Contraction

Lets say for a given positive sequence $(h_n)_n$ and starting points $a_0, a_1>0$ we define the recursive sequence $$ a_{n+1} = \left(1- \frac{h_n}{\sqrt{\sum_{k=0}^{n-1}a_k^2}}\right)a_n $$ Can we ...
Felix B.'s user avatar
  • 2,395
0 votes
0 answers
41 views

Proving that I-cA is a contraction for suitable c

This is probably trivial, but a lot of time has passed since I studied these topics, so I'm a bit rusty. So, let $\mathbf{A}$ be square matrix in $\mathbb{R}^{n\times n}$. We want to prove that there ...
DeltaIV's user avatar
  • 253
4 votes
1 answer
110 views

Using connectedness to prove surjectivity...

$\textbf{My problem:}$ Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a contraction and $\phi :\mathbb{R^2} \rightarrow \mathbb{R^2}$ defined by $\phi (x,y)=(x+f(y),y+f(x))$. Prove that $\phi (\mathbb{R^...
3435's user avatar
  • 385
2 votes
1 answer
60 views

Norm of the trimming map

Let $M_n$ denote the linear space of $n\times n$ matrices over $\mathbb{C}$. Consider $M_n$ as a normed space with operator norm. Is it true that the trimming map $$ T:M_n\to M_n: \begin{pmatrix} a_{1,...
Norbert's user avatar
  • 57k
1 vote
0 answers
50 views

Contraction proof of averaging operator: $\operatorname{Var}_\nu (M f) \leq (1 - \kappa)^2 \operatorname{Var}_\nu (f)$

Let $(X, d, m)$ be an ergodic random walk on a metric space, with invariant distribution $\nu$. Suppose that the coarse Ricci curvature of $X$ is at least $\kappa > 0$ and that the average ...
abc's user avatar
  • 33
1 vote
0 answers
54 views

why contraction mapping theorem fails in this case?

Suppose $f(x) = \cos x$, then for $x \in [0, \frac{\pi}{3}]$, we know that there exists a unique fixed point as the mapping $f$ is contraction mapping on $[0, \frac{\pi}{3}]$. One can see this in plot ...
user808843's user avatar
1 vote
1 answer
178 views

Positive-definite non-symmetric matrix contraction operator?

Let A be positive definite and not symmetric (edit: and real). Why is $I - \alpha A$ a contraction for sufficiently small $\alpha$? I see why this is the case if A is symmetric since it will have an ...
Robin Carter's user avatar
2 votes
1 answer
520 views

Contraction of Bellman Operator under general $L_p$ norms

We know that the Bellman Operator $$ TV(s) = \max_a r(s,a) + \sum_{s' \in S}p(s'|s,a)V(s') $$ is a contraction under $L_\infty$ norm.For reference one can see the following link Proof that Bellman ...
rostader's user avatar
  • 479
1 vote
1 answer
73 views

Fixed Point Iterations on $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$

Say I have two nonlinear equations of the form $$ \begin{bmatrix} u \\ v \end{bmatrix} = f(u,v) = \begin{bmatrix} f_1(u,v) \\ f_2(u,v) \end{bmatrix}, \tag*{(1)} $$ where $u,v \in \mathbb{R}$ and I ...
user594147's user avatar
0 votes
1 answer
125 views

"Expansion" mapping on a compact

Let $(K,d)$ be a compact metric space. We consider $f: K -> K$ : $$\forall x,y \in K: d(x,y) \leq d(f(x), f(y))$$ Show that $$\forall \epsilon >0 :d(f(x), f(y)) \leq d(x, y)+\epsilon$$ What I'...
SilverLight's user avatar
2 votes
0 answers
44 views

Prove that T is not a contraction. [duplicate]

Let $(C[0, 1], d_∞)$ be the metric space of continuous functions on $[0, 1]$ , where the distance function is defined by $$d_∞(f,g)= \sup_{x\in [0,1]}|f(x)-g(x)|$$ Consider the function $T:(C[0, 1],...
john's user avatar
  • 1,288
0 votes
1 answer
106 views

Prove the existence of the limit using fixed point theorem

Consider the sequence $x_{n+1}=x_n-\frac{x_n^2-a}{2}$, prove that $\forall a \in (0,1) $ the sequence converges to $\sqrt{a}$. Considering a function $f(x)=x-\frac{x^2-a}{2}$ we have $$f'(x) = 1 - x,$$...
toss's user avatar
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