The discussion revolves around the continuous extension of a homomorphism defined on polynomials in a bounded normal operator and its adjoint. Participants explore the implications of this extension, particularly focusing on the topology involved in the spaces of bounded operators and polynomials.
Participants express differing views on the definitions and implications of the topologies involved, and the discussion remains unresolved regarding the specifics of the continuous extension and the nature of the mapping.
Limitations include the lack of consensus on the definitions of the topologies in the relevant spaces and the implications of the continuity of the homomorphism.
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).