- (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.

Wed Jul 3 10:27:42 PDT 1996