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### A few polynomial equations

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