When I first considered writing a book about multipliers, it was my intention to produce a moderate sized monograph which covered the theory as a whole and which would be accessible and readable to anyone with a basic knowledge of functional and harmonic analysis. I soon realized, however, that such a goal could not be attained. This realization is apparent in the preface to the preliminary version of the present work which was published in the Springer Lecture Notes in Mathematics, Volume 105, and is even more acute now, after the revision, expansion and emendation of that manuscript needed to produce the present volume. Consequently, as before, the treatment given in the following pages is eclectric rather than definitive. The choice and presentation of the topics is certainly not unique, and reflects both my personal preferences and inadequacies, as well as the necessity of restricting the book to a reasonable size. Throughout I have given special emphasis to the func tional analytic aspects of the characterization problem for multipliers, and have, generally, only presented the commutative version of the theory. I have also, hopefully, provided too many details for the reader rather than too few.

Inhalt
0. Prologue: The Multipliers for L1(G).- 0.0. Introduction.- 0.1. Multipliers for L1(G).- 0.2. Notation.- 0.3. Notes.- 1. The General Theory of Multipliers.- 1.0. Introduction.- 1.1. Elementary Theory of Multipliers.- 1.2. Characterizations of Multipliers.- 1.3. An Application: Multiplications which Preserve the Regular Maximal Ideals.- 1.4. Maximal Ideal Spaces.- 1.5. Integral Representations of Multipliers.- 1.6. Isometric Multipliers.- 1.7. Multipliers and Dual Spaces.- 1.8. The Derived Algebra.- 1.9. The Derived Algebra for Lp(G), 1? pM(Lp(G), Lq(G)), l?p, q??.- 5.5. Some Results Concerning Lp(G)^ and M(Lp(G), Lq(G))^.- 5.6. M(Lp(G), Lq(G)) as a Dual Space, 1?p, q??.- 5.7. Multipliers with Small Support.- 5.8. Notes.- 6. The Multipliers for Functions with Fourier Transforms in Lp (?).- 6.0. Introduction.- 6.1. The Banach Algebras Ap(G).- 6.2. The Multipliers for Ap(G) as Pseudomeasures.- 6.3. The Multipliers for Ap(G): G Noncompact.- 6.4. The Multipliers for Ap(G): G Compact.- 6.5. Notes.- 7. The Multipliers for the Pair (Hp(G), Hq (G)), 1?p, q??.- 7.0. Introduction.- 7.1. General Properties of M(Hp(G), Hq (G)), 1?p, q??.- 7.2. The Multipliers for the Pair (Hp(G), Hq (G)), 1?q?2?p??.- 7.3. The Multipliers for the Pair (Hp(G), H?(G)), 1?p??.- 7.4. Notes.- Appendices.- Appendix A: Topology.- Appendix B: Topological Groups.- Appendix C: Measure and Integration.- Appendix D: Functional Analysis.- Appendix E: Banach Algebras.- Appendix F: Harmonic Analysis.- Author and Subject Index.