Best Fredholm perturbation theorems

Abstract

. Contents of volume XC, number 3 M. SCI-IECHTER and R. W1-nmzv, Best Fredholm perturbation theorems . . L. DREWNOWSKI, On uncountable unconditional bases in Banach spaces . 191-195 P. J. MANGHENI, .?,,-Spaces and cone summing operators. . . . . . . . 197.203 W. ZBLAZKO, Algebraic generation of B(X) by two subalgebras with square zero . . 205-212 M. SKWARCZYNSKI, L‘-Angles between one-dimensional tubes . . 213.233 Contents of volumes LXXXI—XC . . . 235_257 1 75~190 STUDIA MATHEMATICA Managing Editors: Z. Cieslelski, W. Orlicz (Editor-in-Chief), A. Pelczytiski, W. Zelnzko ' The journal prints original papers in English, French, German and Russian, mainly on functional analysis, abstract methods of mathematical analysis and on the theory of probabilities. Usually ’3 issues constitute a volume. The papers submitted should be typed on one side only and accompanied by abstracts, normally not exceeding 200 words. The authors are requested to send two copies, one of them being the typed, not Xerox copy. Authors are advised to retain a copy of the paper submitted for publication. Manuscripts and the correspondence concerning editorial work should be addressed to STUDIA MATHEMATICA ul. Sniadeckich 8, 00—950 Warszawa, Poland Correspondence concerning exchange should be addressed to INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES ul. Sniadeckich 8, 00-950 Warszawa, Poland The journal is available at your bookseller or at ARS POLONA ‘ Krakowskie Przedmiescie 7, 00-068 Warszawa, Poland (9 C°Pl61'1Eh* bl’ P3T'1StW0W6 Wydawnictwo Naukowe, Warszawn 1988 ISBN 33-01-08330-8 ISSN 0039-3223 PRINTED IN POLAND W R0 C LA W.5KI’‘t 53 1‘.i’lil.1q’*ii:’VNs.e1lA_N A U K o'w A STUDIA MATHEMATICA, T. xc. (1988) M Best Fredholm perturbation theorems by M. SCH EC HTE R and R O BERT WH ITLEY (Irvine, Cal.) Abstract. Consider a basic Fredholm perturbation theorem; for example: Let T be a Fredholm operator and suppose that B is a linear operator with |[B|[ l 'y('1q). Then T+B is a Fredholm operator with the same index as 73 The question we consider, for this theorem and others like it, is to what extent can the functions ||B||, v(T), and ||B||/y(T) be improved and still have the conclusions of the theorem hold. I. Basic theorems. The problem of finding the “best possible” constants in perturbation theorems has been studied by many authors. In this first section it is shown that for two classical perturbation theorems it is possible to describe the “best” perturbation constant for that theorem. These specific results raise the question dealt with in the rest of the paper: What is the general form of a perturbation theorem and what constitutes a best theorem. One concrete instance of the problem of finding a best perturbation constant is that of finding the best constant y(T) for the classical Theorem A below; this question is raised in [8, p. 96] for a weaker version of Theo- rem A. The answer to this question is given as Theorem 1 below, and was given in [11] for operators with a bounded inverse. Here, and in what follows, all operators are bounded linear operators mapping a Banach space»X into a Banach space Y, ct( T) is the dimension of the null space N(T) of T, and B(T) is the codimension of the range of '1? A Fredholm operator T has both a(fl and fl(T) finite, and its index is x(T) = oc(T)-fi(T). The perturbation constant y(T) used in [8] depends upon the choice of complementary summands. We will instead here let y(T) denote Kato’s minimum modulus [10, p. 231 or 6, p. 96] defined by (2) y(T) = inf{||'I3c||/d(x, N(T)): d(x, N(T)) g 0}. The operator T has closed range ilf ;:(T) g 0 [6, p. 98]. The 45., operators of Gohberg and Krein [7] are defined by: an operator T is a ¢+ operator if it has closed range and finite nullity at(T). THEOREM A. For any d5+ operator 7: if B is a linear operator with |]B|| l ):(T), then T+B is a Q, operator with: (i) x(T+B) = %(T)- (ii) a(T+B) l at?)-