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Approximation to inverses of normal operators

Published online by Cambridge University Press:  14 November 2011

J. Mika  and
D. C. Pack
J. Mika
Affiliation:
Department of Mathematics, University of Strathclyde, Livingstone Tower, Richmond Street, Glasgow, Gl 1XH, U.K.
D. C. Pack
Affiliation:
Department of Mathematics, University of Strathclyde, Livingstone Tower, Richmond Street, Glasgow, Gl 1XH, U.K.
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Synopsis

For many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.

For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes of operator.

The investigation is extended to the representation of A−1 by a polynomial in A withcomplex coefficients. For polynomials of first and higher orders the application is restricted to bounded self-adjoint operators; at first order, explicit formulae are given.

Application of the method to Fredholm operators is considered in detail, and its use for the point-wise solution to Fredholm integral equations is illustrated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 335 - 345
Copyright
Copyright © Royal Society of Edinburgh 1986

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