Disjointness Preserving Operators and the Navy

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Disjointness Preserving Operators and the Navy

• Or,
• The Matrix Unloaded

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presented by
Gerard
Buskes
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• In what follows, I am not claiming, in any way,
• to have a method that immediately applies
• to the sailor-job situation.
• Goals
• Observe how the very idea of sailor-job matrices
• leads to very natural investigations, which,
• when placed in the context of operator theory
  lead to
• central questions.
• 2. Ask for your help to apply the operator ideas
• to the sailor-job problem and point out that some
  
• applications may present themselves immediately.

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Jobs and Sailors

• The most primitive form of matching jobs to
  sailors
• Each sailor gets to make one choice only. If
  sailor i chooses job j we put a 1 in a big square
  matrix on spot (i,j). In row i we put zeroes
  everywhere else. The resulting matrix is what is
  called a Boolean matrix.
• For the moment we will call the matching
• matrices sailor-job matrices.

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JOBS
SAILORS
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Every one of the Boolean sailor-job matrices is a
so-called stochastic matrix.
A stochastic matrix is one where the numbers
in each row add up to 1. This is a much more
reasonable sailor-job matching process of course.
The results that I will mention later are
presently not known for stochastic matrices.
By building the matching matrix in time, where
later sailors get to see what stochastic choices
their predecessors made, the result will be a
doubly-stochastic matrix.
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• There are many ways to look at a (Boolean)
  matrix.
• One of the interesting things that one can do
  with
• a Boolean matrix is multiply it with another
• Boolean matrix.
• If you multiply a sailor-job Boolean matrix with
  any
• other Boolean matrix you get a Boolean matrix.
• If you multiply two sailor-job Boolean matrices
  then
• you obtain another sailor-job Boolean matrix. In
  other
• words, the sailor-job Boolean matrices are closed
  
• under multiplication, it is a semigroup.

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The sailor-job problem in this terminology Can
one find a scheme that produces a sequence
of sailor-job Boolean matrices that becomes
stable, i.e. that, essentially, turns into a
permutation matrix? The easiest way to make a
sequence of sailor-job matrices from a given one
is to calculate its powers.
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It is time for some calculations.
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Conclusion The matrix
is diagonal when restricted to its range. In
other words, the sequence of powers of A becomes
stable.
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From an operator point of view the matrix
represents an operator from n-dimensional space
to n-dimensional space. The elements, vectors,
of n-dimensional space can be seen as functions
on the set 1,2,..,n. The standard basis of
the domain space are the coordinate vectors,
whose supports are pairwise disjoint, when seen
as functions. A sailor-job Boolean matrix
transforms the standard basis into vectors,
which, when also seen as functions on the set
1,2,..,n, are pairwise disjoint. Thus a
sailor-job matrix represents a disjointness preser
ving operator.
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Why matrices are important for operator theory.
Finite rank operators
Compact operators
matrices
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If one replaces in sailor-job matrices the ones
by other numbers, even negative, or, if you wish,
complex numbers, the conclusion remains
valid For every disjointness preserving matrix
A some power of the matrix is essentially (i.e.
when restricted to the range of some other power
of A) a diagonal matrix, hence stable.
(Boulabiar, Sirotkin, Buskes, 2003)
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Unknown at the moment
is diagonal, but what is the smallest n?
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The previous result remains valid, in all vector
spaces in which one can suitably
define disjointness (like for instance, vector
spaces of functions). Then of course one can not
use matrix techniques because the spaces may not
have finite dimension. The operators have to be
what is called bounded and need to satisfy a
polynomial equation, just like matrices satisfy
their own characteristic equation. The study that
results is of order bounded, disjointness
preserving, algebraic operators.
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From an operator point of view we can use this to
do something one cannot do with compact operators
nor with finite rank operators The absolute
value of order bounded, disjointness preserving,
algebraic operators is algebraic as well (and
order bounded and disjointness preserving.
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• What can the operator approach do for the
  sailor-job situation?
• It simplifies calculating eigenvalues for
  disjointness preserving matrices.
• It would be nice if something similar could
  be said for stochastic or doubly
• stochastic matrices. The Perron-Frobenius
  Theorem does always help.
• 2. Help.

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References()

• K. Boulabiar, G. Buskes After the determinants
  are down a criterion for
• invertibility, to appear in American Mathematical
  Monthly, September 2003.
• 2. K. Boulabiar, G. Buskes Polar decomposition of
  order bounded disjointness
• preserving operators, to appear in Proceedings
  AMS 2003.
• 3. Melvin Henriksen, Karim Boulabiar, and G.
  Buskes A generalization of a Theorem
• on Biseparating maps, Journal of Mathematical
  Analysis and Applications, 280,
• (2003), 334-349.
• 4. Diagonalizable order bounded disjointness
  preserving operators, submitted to
• Integral Equations and Operator Theory, 2003,
  with K. Boulabiar and G. Sirotkin.

() Supported by Office of Naval research Grant
N00014-01-1-0322