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Nonselfadjoint Operators and Related Topics

Workshop on Operator Theory and Its Applications, Beersheva, February 24–28, 1992

  • Editors
  • A. Feintuch
  • I. Gohberg

Part of the Operator Theory: Advances and Applications book series (OT, volume 73)

Table of contents

  1. Front Matter
    Pages I-X
  2. Daniel Alpay, Laurent Baratchart, Andrea Gombani
    Pages 30-66
  3. Asher Ben-Artzi, Israel Gohberg
    Pages 96-126
  4. Rajendra Bhatia, Chandler Davis
    Pages 127-137
  5. Gene Christner, Kin Y. Li, James Rovnyak
    Pages 138-181
  6. Peter Fillmore, Masoud Khalkhali
    Pages 256-263
  7. Eric Nordgren, Peter Rosenthal
    Pages 362-368
  8. Back Matter
    Pages 417-422

About these proceedings

Introduction

Our goal is to find Grabner bases for polynomials in four different sets of expressions: 1 x- , (1 - x)-1 (RESOL) X, 1 x- (1 - xy)-1 (EB) X, , y-1, (1-yx)-1 y, (1_y)-1 (1-x)-1 (preNF) (EB) plus and (1 - xy)1/2 (1 - yx )1/2 (NF) (preNF) plus and Most formulas in the theory of the Nagy-Foias operator model [NF] are polynomials in these expressions where x = T and y = T*. Complicated polynomials can often be simplified by applying "replacement rules". For example, the polynomial (1 - xy)-2 - 2xy(1-xy)-2 + xy2 (1 - xy)-2 -1 simplifies to O. This can be seen by three applications of the replacement rule (1-xy) -1 xy -t (1 - xy)-1 -1 which is true because of the definition of (1-xy)-1. A replacement rule consists of a left hand side (LHS) and a right hand side (RHS). The LHS will always be a monomial. The RHS will be a polynomial whose terms are "simpler" (in a sense to be made precise) than the LHS. An expression is reduced by repeatedly replacing any occurrence of a LHS by the corresponding RHS. The monomials will be well-ordered, so the reduction procedure will terminate after finitely many steps. Our aim is to provide a list of substitution rules for the classes of expressions above. These rules, when implemented on a computer, provide an efficient automatic simplification process. We discuss and define the ordering on monomials later.

Keywords

operator operator theory systems theory

Bibliographic information

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