Tagged Questions

40 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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Finding inverse of a general linear transform

I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification. Let's define a general linear transform as $$\int_XK(\mathbf{\omega},x)f(x)dx$$ where $X$ is some ...
34 views

Blind deconvolution of a function convolved with itself

I have a function/vector $f$ that I know is the result of an unknown function $g$ convolved with itself: $f = g \ast g$ Is there any way to do a blind deconvolution on $f$ with this constraint?
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Integral of inverse function

On wikipedia and on the following mathstackexchange page, a formula for the sum of the integrals of a function and its inverse (with "corresponding" limits) is given, do you have a proper proof for ...
92 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$(I-T)^{-1}=\sum_{n=0}^\infty T^n.$$ For the existence of $(I-T)^{-1}$ ...
34 views

Inverse function without the original function

I am going through this paper, 'Certifiable Quantum Dice Or, True Random Number Generation Secure Against Quantum Adversaries' by Vazirani and Vidick. In 'Our results' section on the page 2, it says: ...
247 views

Invertibility of a linear operator on a Hilbert space.

Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such ...
221 views

Bounded operator inverse, norm and spectrum

I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator: $X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$ 1. I need to show that an ...
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If $A$ is invertible, so is $A^*A$

Let $A \in L(H)$, for a Hilbert space $H$. If $A$ is invertible, why is $A^*A$ invertible, too?
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Inverse of inverse of function?

What is the inverse of inverse of a function (I assume the original function is invertible)? Is this the original function? Is it always true?
For $T$ compact, $I-T$ left or right invertible implies $I-T$ invertible
Let $S\in B(X)$ be a bounded linear operator from $X$ onto $X$ and let $T\in K(X)$ be a compact linear operator from $X$ onto $X$. Then $$S(I-T)=I \iff (I-T)S=I.$$ I don't know if we need the fact ...