What have you tried so far? When you use words like "continuously" and "closure", you are implying the existence of a topology in the space B(H) and also in the space of polynomials. How do you define these topologies?Boromir said:Let $T$ be a bounded normal operator and let $x$ be a member of the spectrum. Consider the homomorphism defined on the set of polynomials in $T$ and $T^{*}$ given by $h(p(T,T^*))=p(x,x^*)$ Prove that this map can be continuosly extended to the closure of $P(T,T^*)$
Opalg said:What have you tried so far? When you use words like "continuously" and "closure", you are implying the existence of a topology in the space B(H) and also in the space of polynomials. How do you define these topologies?
Okay, it's the operator norm in B(H). But what norm are you using on the space of polynomials?Boromir said:just the usual one given by the operator norm. I can see that I need to show 2 things, namely that lim$p_{n}(x,x^*)$ exist given that the corresponding sequence in B(H) converges, and that the map is well defined.
Opalg said:Okay, it's the operator norm in B(H). But what norm are you using on the space of polynomials?
The range of the mapping $h$ consists of expressions of the form $p(x,x^*)$. What is $x$ supposed to mean there, and what is $p(x,x^*)$? (It's not an operator).Boromir said:a polynomial of operators is just an operator so the same norm.
Opalg said:The range of the mapping $h$ consists of expressions of the form $p(x,x^*)$. What is $x$ supposed to mean there, and what is $p(x,x^*)$? (It's not an operator).
Continuous extension of homomorphism is a mathematical concept that involves extending a homomorphism, which is a function that preserves algebraic structure, to a larger domain while maintaining its properties.
Continuous extension of homomorphism is important because it allows for the application of algebraic concepts and techniques to a wider range of mathematical structures, making it a useful tool in various fields such as topology, functional analysis, and abstract algebra.
A continuous extension of homomorphism must preserve the algebraic operations of the original homomorphism and also maintain continuity, meaning that small changes in the input should result in small changes in the output.
Continuous extension of homomorphism is closely related to topology because it relies on the concept of continuity, which is a fundamental concept in topology. In fact, continuous extension of homomorphism is often used in topology to prove certain theorems and properties.
Yes, continuous extension of homomorphism has various real-world applications, such as in signal processing, image reconstruction, and data analysis. It is also used in physics and engineering to model and analyze continuous systems.