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 ...
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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 ...
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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 ...
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1 vote
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$\forall\ x,y\in [a,b], |f(x)-f(y)|<|x-y|$ 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 ...
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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....
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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 ...
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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 ...
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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 ...
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