How Does Finite Dimensionality Affect the Spectrum in a Banach Algebra?

Tien
Messages
1
Reaction score
0

Homework Statement



How to show element of finite dimensional banach algebra has finite spectrum?

Homework Equations



spectrum(x) = set of complex numbers 'c' with cI-x not invertible, I is identity

The Attempt at a Solution



please help to start, I don't know
 
Physics news on Phys.org
According to the rules of this forum, you have to show an effort at solving this before we can provide help.
 
Try showing that the eigenvalues belonging to different elements of the spectrum are linearly independent.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
View full post »

Similar threads

Show that a space is a Banach space
Replies
12
Views
2K
Homeomorphism in a Banach space
Replies
16
Views
2K
A quite verbal proof that if V is finite dimensional then S is also....
Replies
34
Views
3K
Dimension statement about (finite-dimensional) subspaces
Replies
15
Views
1K
Sum of two closed subspaces in a Banach space
Replies
6
Views
3K
Spectrum of Invertible Elements in Unital Banach Algebra: A Proof
Replies
4
Views
2K
Maximum norm and Banach fixed-point theorem
Replies
9
Views
2K
Does the compact subset of an infinite Banach have finite span?
Replies
6
Views
3K
Distance between fixed points of contracting maps
Replies
12
Views
2K
If A is a Banach algebra, then so is A/I
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top