I Parameterization of linear operators on the holomorphisms

Whiteboard_Warrior
Messages
2
Reaction score
0
Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear maps on the space of holomorphisms? If so, could someone provide a proof of this? If not, what are some counterexamples?

P.S. I promise I'll post more interesting questions in the future :)
 
Physics news on Phys.org
bump?
 

Thread 'Volterra equation of the second kind'

There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
View full post »

Thread 'Visual Representation of Separation of Varables'

Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
View full post »

Similar threads

I Proving a basis is a basis for homogeneous linear differential equation with constant (complex) coefficients
Replies
2
Views
2K
I Second order non-homogeneous linear ordinary differential equation
Replies
2
Views
3K
I Why is this differential equation non-linear?
Replies
12
Views
2K
I Why do ##t## and ##-i\hbar\partial_t## not satisfy the definition of a linear map/operator in Hilbert space?
Replies
4
Views
2K
I If T is diagonalizable then is restriction operator diagonalizable?
Replies
3
Views
2K
A What happens when an operator maps a vector out of the Hilbert space?
2
Replies
59
Views
4K
I Chebyshev Differentiation Matrix
Replies
2
Views
2K
I Lindbladian Jump Operators Condition
Replies
0
Views
2K
I Proof by induction of block diagonal decomposition of a matrix
Replies
4
Views
1K
I Differential operator vs one-form (covector field)
Replies
10
Views
3K

Hot Threads

Recent Insights

Back
Top