Variation of the Spectrum of Operators in Infinite Dimensional Spaces

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.


Introduction
Let X be an infinite dimensional Banach space.We denote by an arbitrary bounded operator on T X and by I the identity operator on X .Let be the closed unit disc of the complex plane .The restriction on of the spectrum of an operator , denoted by , is the unit spectrum defined as follows:


Essential spectra of some matrix operators on Banach spaces (see [1]) and spectra of some block operator matrices (see [2]) were investigated, with applications to differential and transport operators.This paper investigates the variations of the unit spectrum as T varies over the space of all bounded operator on the Banach space   T    L X X .First, we introduce the sets and the topologies required for this study.

Definition 1   
  the set of all compact subsets of the closed unit disc  of the complex plane  ; The set is endowed with the Hausdorff topology generated by the families of all subsets in one of the following forms for an open subset of  .Therefore, V

 
  is a Polish space, i.e., a separable metrizable complete space, since is Polish (see [3][4][5]).It is shown below that we can reduce the families that generate the above Hausdorff topology.

 
  be the set of compact subsets of the closed unit disc  .Then

 
  equipped with the Hausdorff topology is a Polish space; where the Borel structure is generated by one of the following two families

Proof 1 Let be an open subset of . There exists a decreasing sequence of open subsets
, and since is decreasing, we have

Norm Operator Topology and the Spectrum Function
We equip with the canonical norm of operators defined by is not continuous when is endowed with its canonical norm.
  converge to the identity I while and .However, we have the following result.

 
 the space of bounded operators equipped with the norm of operators, and the set of compact subsets of the unit disc equipped with the Hausdorff topology.Then the spectrum map , then for all is continuous (see [6]).
For any , .L X .

Strong Operator Topology and the Spectrum Function
Consider now   L X equipped with the strong operator topology op (see [6]).In general, S

 
L X equipped with the strong operator topology is not a polish space (since it is not a Baire space).However, if X is separable, then is a standard Borel space.Indeed, it is Borel-isomorph to a Borel subset of the Polish space X  equipped with the norm product topology via the map The next result shows how this topology on   L x affects the spectrum function.
Theorem 1 For any separable infinite dimensional Banach X , the map By a descriptive set theory result from ( [7]), to show Copyright © 2013 SciRes.APM that V is a Borel set it suffices to show that E  is a Borel set with K  vertical sections.For , the vertical section of the set along the direction T is given by Hence, to finish the proof, it is enough to prove the following claim. Claim: is not isomorph to its range , is not dense in .
is an isomorphism onto its range, then is a closed subspace that will be strict if , and thus not dense in On the other hand, since X is separable, there exists a countable and dense subset in the sphere of  X S X , and there exists a dense sequence   n n x  in X .Now, we will show that A and are Borel sets.

Let
. From the definition of .
is equipped with the the strong operator convergence op , it follows that the sets x k A are open.Hence, A is a Borel set.
On the other hand, "    or again,   and 1 such that 1 : .
are Borel sets.Hence is also a Borel set.This proves the claim and ends the proof of the theorem 1.

Conclusions
The variation of the unit spectrum of operators in infinite dimensional Banach spaces is investigated.The unit spectrum of an operator , denoted by , is defined as the restriction on the closed unit disc of the complex plane  of the spectrum of given by

 
L X is endowed with the norm of operators.On the other hand, when   L X is endowed with the strong operator topology, it is shown that first X needed to be a separable infinite dimensional Banach to guarantee a standard Borel structure on , then it is shown that the that the map is Borel in this case.Therefore, this topology is making the spectrum function more rigourous, and as a consequence the variations of the spectrum following changes in an operator or a se-

B
paper presents a simplified characterization of the Borel structure making the set of compact subsets of the closed unit disc a Polish space.It is also shown that for a Banach space