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We set the stage by listing a (very) few polynomial equations which
commonly occur in analysis.
- (1)
a matrix T is an isometry if and only if
T*T - 1 = 0,
that is,
if
(T, T*)
is a zero of a (noncommutative) polynomial.
- (2)
X satisfies a Ricatti equation
A X + X A* +
X R X + Q = 0
if and only if
(A, A*, X, R, Q)
is a zero of a (noncommutative) polynomial.
- (3)
a matrix T is an coisometry if and only if , that is,
if is a zero of a (noncommutative) polynomial.
- (4)
a matrix A is invertible if and only if there exists a matrix
B such that AB-1=0 and BA-1=0, that is, if (A,B)
is a common zero of two (noncommutative) polynomials.
- (5)
a bounded linear transformation V of a complex Hilbert
space is a partial isometry if and only if ,
that is, is a zero of a (noncommutative) polynomial.
- (6)
a bounded linear transformation T of a complex Hilbert
space has a pair of complimentary invariant subspaces
if and only if there exist operators and
such that
, ,
, ,
,
and
, that is,
is a
common zero of seven (noncommutative) polynomials.
- (7)
using a polar decomposition of a bounded linear
transformation of a complex Hilbert space T is equivalent
to introducing bounded linear
transformations P and V such that T= PV,
, and ,
that is, is a common zero of three
(noncommutative) polynomials and one
additional constraint holds. Using the idea of a
strategy (which is to be given in this paper), one
can use such additional
constraints to one's advantage during an algebraic
computation.
- (8)
is a pair of commuting isometries if and only
if and are isometries ( , )
and and commute ( ).
The pair lifts to a pair of commuting unitaries if
and only if there exists , , and such
that and are an isometries ( ,
), and are unitaries
( , , ,
) and and commute ( )
and intertwines and for j=1,2
( and ).
Therefore, the pair lifts to a pair of commuting unitaries
if and only if
is a common zero of 12
(noncommutative) polynomials.
We can summarize the above examples by saying that a number of
properties of matrices and operators (bounded linear transformations
of complex Hilbert space) are equivalent to the statement that
a tuple of matrices (or tuple of operators)
is a common zero of a set of
polynomial equations. We now discuss the traditional
connection between common
zeros and ideals.
Next: Solutions of Polynomial Equations
Up: Background on polynomial equations
Previous: Background on polynomial equations
Helton
Wed Jul 3 10:27:42 PDT 1996