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Title Nonselfadjoint Operators and Related Topics [electronic resource] : Workshop on Operator Theory and Its Applications, Beersheva, February 24-28, 1992 / edited by A. Feintuch, I. Gohberg Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1994 http://dx.doi.org/10.1007/978-3-0348-8522-5 X, 422 p. online resource

SUMMARY

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

CONTENT

Joint spectrum and discriminant varieties of commuting nonselfadjoint operators -- 1. Introduction -- 2. Joint spectra of commuting operators with compact imaginary parts -- 3. Colligations and vessels -- 4. The discriminant varieties -- References -- On the differential structure of matrix-valued rational inner functions -- 1. Introduction and preliminaries -- 2. The differential structure of Inp -- 3. Charts using Schur algorithm -- 4. Conclusion -- References -- Conservative dynamical systems and nonlinear Livsic-Brodskii nodes -- 1. Conservative systems -- 2. Nonlinear Livsic-Brodskii nodes: models for a given dynamics up to energy preserving diffeomorphic change of variable -- 3. Other partionings of the cast of characters into knowns and unknowns -- References -- Orthogonal polynomials over Hilbert modules -- 1. Introduction -- 2. Orthogonalization with invertible squares -- 3. Preliminaries on inertia theorems for unilateral shifts -- 4. The main result -- References -- Relations of linking and duality between symmetric gauge functions -- 1. Introduction -- 2. Linked symmetric gauge functions -- 3. Quotient of symmetric gauge functions -- 4. Q-norms -- References -- Julia operators and coefficient problems -- 1. Introduction -- 2. Julia operators for triangular matrices -- 3. Multiplication transformations on power series -- 4. Extension problem for substitution transformations -- Appendix. Formal algebra -- References -- Shifts, realizations and interpolation, Redux -- 1. Introduction -- 2. Formulas and facts -- 3. R? variance -- 4. Realizations -- 5. Reproducing kernel spaces -- 6. H(S) spaces -- 7. A basic interpolation problem -- 8. Factorization and recursive methods -- 9. Characteristic functions -- References -- Arvesonโ{128}{153}s distance formulae and robust stabilization for linear time-varying systems -- 1. Introduction -- 2. Preliminaries -- 3. Stabilization and proper representations -- 4. Robust stabilization: Proper representation uncertainty -- 5. Gap metric robustness -- Entire cyclic cohomology of Banach algebras -- 1. Background -- 2. Definitions -- 3. Results -- References -- The bounded real characteristic function and Nehari extensions -- 1. Introduction -- 2. Bounded real functions -- 3. Hankel operators -- 4. State space realizations -- 5. Suboptimal Nehari extensions -- References -- On isometric isomorphism between the second dual to the โ{128}{156}smallโ{128}{157} Lipschitz space and the โ{128}{156}bigโ{128}{157} Lipschitz space -- The Kantorovich-Rubinstein norm -- Completion of the space of measures in the KR norm -- Critical and noncritical metric spaces -- References -- Rules for computer simplification of the formulas in operator model theory and linear systems -- I. Introduction -- II. The reduction and basis algorithms -- III. Operator relations with finite basis for rules -- IV. Operator relations with infinite basis for rules -- V. A new algebra containing the functional calculus of operator theory -- VI. Grรถbner basis property -- VII. Summary of practical rules you might use -- References -- Some global properties of fractional-linear transformations -- Preliminaries -- 1. The case of invertible plus-operators -- 2. The general case of a non-invertible operator U -- References -- Boundary values of Berezin symbols -- 1. Introduction -- 2. Compactness criterion -- 3. Continuous Berezin symbols -- 4. Two questions -- References -- Generalized Hermite polynomials and the bose-like oscillator calculus -- 1. Introduction -- 2. Generalized Hermite polynomials -- 3. The generalized Fourier transform -- 4. Generalized translation -- 5. The Bose-like oscillator -- References -- A general theory of sufficient collections of norms with a prescribed semigroup of contractions -- 1. Formulation of the problem -- 2. Notions -- 3. Formulations of results -- 4. Proofs of results -- References

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