What is the definition of analytic at infinity in complex analysis?

  • Context: Graduate 
  • Thread starter Thread starter gauss mouse
  • Start date Start date
  • Tags Tags
    Infinity
Click For Summary

Discussion Overview

The discussion revolves around the concept of "analytic at infinity" in complex analysis, particularly in the context of continuous linear operators in Banach spaces. Participants seek definitions and basic results related to this concept, as well as its relation to "bounded at infinity."

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant requests a definition of "analytic at infinity" and mentions difficulty finding information on the topic.
  • Another participant proposes that a function f(z) is analytic at infinity if f(1/z) is analytic at zero, suggesting a similar definition for "bounded at infinity" but requests more context.
  • A participant clarifies that they are examining two specific Banach spaces: an arbitrary infinite-dimensional Banach space X and the space of all bounded linear operators from X to X.
  • One participant questions the relevance of analytic functions and inquires whether the discussion pertains to defining a "functional calculus" for operators.
  • The original poster confirms that they are indeed trying to define a functional calculus involving analytic functions and bounded operators.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and context regarding the definition of "analytic at infinity." There is no consensus on the complete definition or its implications, as further clarification is sought.

Contextual Notes

Participants note the importance of context in understanding the definitions, particularly regarding the types of Banach spaces and the nature of the analytic functions involved.

gauss mouse
Messages
24
Reaction score
0
Hi, I'm reading "Spectral Theory of Linear Operators" by John Dowson. I've seen the phrase "analytic at infinity" popping up very early in the book, but no definition is given. I wonder if anyone could tell me what the definition is or where I might find the definition and perhaps a few basic results on analyticity at infinity? The operators I'm looking at are continuous linear maps from a complex Banach space to itself, so my question is really about complex analysis.

I've tried Google but have found no definition.

I think "bounded at infinity" is a related concept, but again I do not know what that means. Maybe someone can help me out there too?

Thanks.
 
Physics news on Phys.org
f(z) is analytic at infinity if f(1/z) is analytic at zero. Probably the same kind of definition can be given for "bounded at infinity" - but to be sure, you'll need to give us some context. I guess the Banach spaces you're working with are some kind of function spaces.

Anyway, this kind of stuff should be viewed in the context of doing complex analysis on the Riemann sphere (= the complex plane plus a "point at infinity"). This might help you in your search for appropriate reading material.
 
morphism said:
f(z) is analytic at infinity if f(1/z) is analytic at zero. Probably the same kind of definition can be given for "bounded at infinity" - but to be sure, you'll need to give us some context. I guess the Banach spaces you're working with are some kind of function spaces.
Thanks, I'm looking at two Banach spaces. An arbitrary infinite-dimensional Banach space X and the Banach space of all bounded linear operators from X to X. Is that enough context? I'm not sure what else to say.
 
No, that's not enough context! Where are the analytic functions coming in? Are you trying to define a "functional calculus"? I.e. are you trying to assign meaning to f(T) where f is an analytic function and T is a bounded operator on X?
 
Yes that's exactly what I'm trying to do. Sorry should've said that.
 

Similar threads

Undergrad Why are analyticity and convergence related in complex analysis?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Precise intuition about limits and infinitesimals
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Other Complex analysis or calculus?
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
2K
Courses Complex Analysis Courses or Complex Variable Courses?
  • · Replies 18 ·
Replies
18
Views
4K
Graduate Cauchy Sequences - Complex Analysis
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Why do ##t## and ##-i\hbar\partial_t## not satisfy the definition of a linear map/operator in Hilbert space?
  • · Replies 4 ·
Replies
4
Views
2K