Non-Archimedean Linear Operators and Applications

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N ON -A RCHIMEDEAN L INEAR O PERATORS AND A PPLICATIONS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

N ON -A RCHIMEDEAN L INEAR O PERATORS AND A PPLICATIONS

T OKA D IAGANA

Nova Science Publishers, Inc. New York

c 2009 by Nova Science Publishers, Inc.

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NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data Available upon request ISBN978-1-61470-561-1 (eBook)

Published by Nova Science Publishers, Inc. ✜ New York

In memory of Mabaye Galledou, my grandmother

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Non-Archimedean Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Non-Archimedean Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The t-Vector Space Kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Construction of Q p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 The Field Q p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.3 Convergence of Power Series over Q p . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Construction of K((x)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2

Non-Archimedean Banach and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Non-Archimedean Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Examples of Non-Archimedean Banach Spaces . . . . . . . . . . . . . . . . . . 2.2 Free Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Non-Archimedean Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Non-Archimedean Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Hilbert Space Eω1 × Eω2 × ... × Eωt . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Non-Archimedean Bounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Bounded Linear Operators on Non-Archimedean Banach Spaces . . . . . . . . . 19

11 11 11 12 13 13 13 14 14 14 16 18

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Toka Diagana 3.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Banach Algebra B(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Further Properties of Bounded Linear Operators . . . . . . . . . . . . . . . . . . Bounded Linear Operators on Non-Archimedean Hilbert Spaces . . . . . . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Representation of Bounded Operators By Infinite Matrices . . . . . . . . . 3.3.3 Existence of the Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Examples of Bounded Operators with no Adjoint . . . . . . . . . . . . . . . . . Perturbation of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Further Properties of Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . . 3.5.3 Completely Continuous Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 22 22 25 25 25 26 28 29 31 32 32 38 39 40 40 42 42

4

Non-Archimedean Unbounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Existence of the Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Examples of Unbounded Operators With no Adjoint . . . . . . . . . . . . . . 4.3 Closed Linear Operators on Eω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Diagonal Operators on Eω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 43 44 45 45 46 49 51 51

5

Non-Archimedean Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Continuous Linear Functionals on Eω . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Bounded Bilinear Forms on Eω × Eω . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Unbounded Bilinear Forms on Eω × Eω . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Closed and Closable non-Archimedean Bilinear Forms . . . . . . . . . . . . . . . . . . 5.3.1 Closedness of the Form Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Construction of a non-Archimedean Hilbert Space Using Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 53 53 55 56 58 59

3.3

3.4 3.5

3.6 3.7

61

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5.3.3 Further Properties of the Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4 Representation of Bilinear Forms on Eω × Eω by Linear Operators . . . . . . . . 63 5.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6

Functions of Linear Operators on Eω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Products and Sums of Diagonal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Integer Powers of Diagonal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Functions of Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Functions of Some Symmetric Square Matrices Over Q p × Q p . . . . . . . . . . . 6.5.1 The Powers of the Matrix T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Exponential of the Matrix T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 69 72 73 75 75 76 76

7

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Properties of non-Archimedean C0 -Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Existence of Solutions to Some p-adic Differential Equations . . . . . . . . . . . . . 7.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 79 83 86 86

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Preface This self-contained book provides the reader with a comprehensive presentation of the author’s recent investigations on operator theory over non-Archimedean Banach and Hilbert spaces. It therefore completes topics uncovered in the author’s recent book, “An Introduction to Classical and p-adic Operator Theory and Applications”, which has recently been published by Nova Science Publishers. Moreover, it offers several new research lines, which are, in most part, inspired by their classical counterparts.

Chapter 1 is devoted to basic tools related to (complete) non-Archimedean valued fields needed in the sequel. Among other things, the ideas of construction of the fields of p-adic numbers and that of formal Laurent series are discussed. Many key results on the topology of the field of p-adic numbers are also discussed. A valued field (K, | · |) is called non-Archimedean whether the following stronger (triangle) inequality holds |σ + ς| ≤ max(|σ|, |ς|), for all σ, ς ∈ K with equality whenever |σ| 6= |ς|. The non-Archimedean operator theory consists of studying the analogues of the classical operator theory when the (complete) non-Archimedean field (K, | · |) plays the role usually played by the fields of real or complex numbers. Classical examples of such fields include Q p the field of p-adic numbers (p ≥ 2 prime), C p the field of p-adic complex numbers, and the field of formal Laurent series.

Chapter 2 introduces basic properties of non-Archimedean Banach spaces, free Banach spaces, and non-Archimedean Hilbert spaces needed in the sequel. One should point out that this is a key chapter, especially for beginners. Chapter 3 is entirely devoted to bounded linear operators on non-Archimedean Banach and Hilbert spaces. In particular, one takes a closer look into the existence of an adjoint for a given bounded linear operator with respect to the non-Archimedean “inner product”. Examples of bounded linear operators, which do not have adjoint also are discussed. New

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developments on perturbations of non-Archimedean bases are also investigated. A nontrivial example of an orthogonal base constructed by perturbation of the canonical orthogonal base of the non-Archimedean Hilbert space Eω , is also given. In the middle of this chapter, special attention is paid to Hilbert-Schmidt and completely continuous operators. Some of the results go along the classical line and others deviate from it. For the most part, their classical counterparts inspire the statements of the results we present. However, their proofs may depend heavily on the non-Archimedean nature of Eω and the ground field K. In contrast with the classical context, the definition of a Hilbert-Schmidt operator does depend on the base. Furthermore, the adjoint of Hilbert-Schmidt operator does exists and also is a Hilbert-Schmidt operator, and each Hilbert-Schmidt operator is completely continuous. As in the classical setting, a natural inner product is considered for Hilbert-Schmidt operators. To illustrate abstract results, several examples are discussed at the end of this chapter.

Chapter 4 examines non-Archimedean unbounded linear operators recently introduced in the literature in Diagana [17]. Several examples are discussed including those arising on the free Banach space C(Z p , Q p ) of all continuous functions from the ring of integers, Z p , into Q p , the field of p-adic numbers. Chapter 5 studies bounded and unbounded non-Archimedean bilinear forms. The closure and the closedness of (unbounded) non-Archimedean bilinear forms are investigated. In particular, necessary conditions for the closedness of the form sum of closed nonArchimedean bilinear forms are given. Chapter 6 consists of a preliminary work of the author on functions and fractional powers of linear operators on non-Archimedean Hilbert spaces. This chapter is illustrated by a few examples including functions of a specific (symmetric) square matrix over Q p × Q p.

Chapter 7 provides the reader with a brief and recent conceptualization of semigroups within the non-Archimedean framework due to Diagana[23]. Non-Archimedean semigroups play an important role in the solvability of p-adic differential and partial differential equations as strong (mild) solutions to the Cauchy problem related to several classes of differential and partial differential equations in the classical setting can be expressed through C0 -semigroups. An example of solvability of differential equations using the concept of a non-Archimedean semigroup is discussed at the end of this chapter. “Non-Archimedean Linear Operators and Applications” is aimed at expert readers, young researchers, beginning graduate and advanced undergraduate students, who are interested in non-Archimedean functional analysis and operator theory as well as their wide range of applications. The basic background for the understanding of the material presented is timely provided throughout the text. Furthermore, the book offers several kinds of examples, ranging from routine to challenging. The end of each chapter offers some useful bibliographical notes and open problems.

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Acknowledgements. I would like to express my deepest thanks to colleagues and friends Professors Dariusz Bugajewski and John M. Rassias for proofreading the first versions of the manuscript. I wish to express special thanks to my colleague and friend, Professor Franc¸ois Ramaroson for our frequent discussions on the so-called p-adic operator theory, and support. I am grateful to Dr. Frank Columbus, Ms. Maya Columbus, and Ms. Donna Dennis from Nova Science Publishers, for their strong editorial support and help. I wish to thank the members of our Department of Mathematics as well as the Howard University College of Arts and Sciences for their strong support since I joined Howard University a few years ago. Last but certainly not least, I am grateful to my wife and our wonderful children for their continuous support, their encouragements, and especially for their putting up with me during all those long hours I spent from them, while writing this book.

Washington, D.C.

Toka Diagana June 2006

Chapter 1

Non-Archimedean Valued Fields 1.1 Introduction This chapter provides the reader with the basic tools on the so-called Non-Archimedean valued fields. Examples of those fields include, but not limited to, Q p (p ≥ 2 is prime), the field of p-adic numbers, C p , the field of complex p-adic numbers, and K((x)), the field of formal Laurent series. All these fields satisfy the so-called non-Archimedean (or ultrametric) triangle inequality formally given by |σ + ς| ≤ max(|σ|, |ς|), for all σ, ς ∈ K with equality whenever |σ| 6= |ς|, where | · | is the corresponding absolute value. A valued field, which does not satisfy the latter inequality is called Archimedean. Examples of Archimedean fields include (R, | · |), the field of real numbers equipped with its natural absolute value, (C, | · |), the field of complex numbers equipped with its absolute value, and many others. Using the non-Archimedean triangle inequality, one derives several properties of those non-Archimedean fields. In term of geometry, we will see that in a non-Archimedean field, each ball is both closed and open, the parallelogram law could turn into an inequality1 , every triangle is isosceles, etc. One of the main objectives of this book is to make the (non-Archimedean) material utilized accessible to many people including those who are not familiar with number theory. For that, proofs of the results, which require some advanced algebraic concepts will simply be omitted. This chapter is organized as follows: Section 1.2 examines topologies of nonArchimedean valued fields. Section 1.3 discusses the construction of Q p , the field of p-adic numbers. Section 1.4 introduces K((x)), the so-called field of formal Laurent series. Throughout the rest of the book R, C, Q, Q p , C p , Z p , N, and Z stand for the fields of real, complex, rational, p-adic, p-adic complex numbers, the ring of p-adic integers, the set of natural integers, and the set of all integers, respectively. 1

In a non-Archimedean valued field (K, | · |), the non-Archimedean parallelogram law states that for all x, y ∈ K, |x + y|2 + |x − y|2 ≤ 2 max(|x|2 , |y|2 ).

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1.2 Non-Archimedean Valued Fields This section introduces the so-called non-Archimedean valued fields. Those fields will play a key role throughout the rest of the book. We will especially emphasis on their topological properties, which are mainly driven by the non-Archimedean (or ultrametric) inequality 2 . In most of our examples, we will make extensive use of the field of the p-adic numbers, known as the simplest (complete) non-Archimedean valued field. The literature on the algebraic properties of those non-Archimedean fields is very extensive; here, we omit those details and refer the reader to most good books in non-Archimedean functional analysis, especially Amice[2], Schikhof[64], Gouvˆea[33], Koblitz[42], Mahler[50], Monna[51], van Rooij[62], and many others. 1.2.1 Basic Definitions Definition 1. A field K is said to be non-Archimedean if it is endowed with an absolute value | · | : K 7→ [0, ∞), which satisfies the following conditions: (1) |x| = 0 if and only if x = 0; (2) |x y| = |x| . |y|, for all x, y ∈ K; and (3) |x + y| ≤ max(|x|, |y|) for all x, y ∈ K. As it had previously been mentioned, the property [Definition 1.3.1, (3)], is called the non-Archimedean triangle inequality. An immediate consequence of the non-Archimedean triangle inequality is the following: (4) |x + y| = max(|x|, |y|) whenever |x| = 6 |y|, where x, y ∈ K. As in the classical setting, if (K, | · |) is a non-Archimedean field, then its absolute value | · | induces a metric d on K × K defined by d(u, v) = |u − v|, u, v ∈ K. The metric d enables to consider the notion of convergence in K. Basically, one says that the sequence (ut )t∈N ⊂ K converges to u ∈ K whether d(ut , u) = |ut − u| → 0 as t → ∞. Now if the metric space (K, d) is complete3 , one says that (K, | · |) is a complete nonArchimedean field. Otherwise, one can always complete it and denote its completion by ˜ | · |). It can be easily shown that (K, ˜ | · |) is also a non-Archimedean field, which obvi(K, ously is complete. In contract with the classical setting, in a non-Archimedean field, every triangle is isosceles. This obviously is a consequence of the non-Archimedean triangle inequality. 2 3

Throughout the book, the word “non-Archimedean” will be preferred to that of “ultrametric”. A metric space (K, d) is said to be complete if every cauchy sequence of elements in K converges. Note that in the non-Archimedean setting, a sequence (un )n∈N ⊂ (K, d) is a Cauchy sequence if and only if d(un+1 , un ) → 0 as n → ∞, see Proposition2.

Non-Archimedean Valued Fields

3

Proposition 1. Let (K, | · |) be (metric space) a non-Archimedean field and let x, y, z ∈ K such that |x − z| = 6 |z − y|. Then |x − y| = max(|x − z|, |z − y|). Proof. Suppose that |x − z| < |z − y|. (The proof for the case |z − y| < |x − z| follows along the same line.) It follows that |x − y| ≤ max(|x − z|, |z − y|) = |z − y|. Suppose that |x − y| < |z − y|. Consequently, |z − y| ≤ max(|z − x|, |x − y|) < |z − y|, which obviously is in contradiction with the fact |x − y| < |z − y|, by assumption, and therefore, |x − y| = max(|x − z|, |z − y|) = |z − y|. Definition 2. Let (K, | · |) be a non-Archimedean field. A sequence (xt )t∈N of K is said to converge to x if for each ε > 0 there exists t0 ∈ N such that |xt − x| < ε whenever t ≥ t0 . A sequence (xt )t∈N of K is called a Cauchy sequence if for each ε > 0 there exists t0 ∈ N such that |xt − xs | < ε whenever s ≥ t ≥ t0 . In contrast with the classical, the following useful result, which also is a consequence of the non-Archimedean inequality, holds: Proposition 2. Let (K, | · |) be a non-Archimedean field. A sequence (xt )t∈N ⊂ K is a Cauchy sequence if and only if |xt+1 − xt | → 0 as t → ∞. Proof. Let (xt )t∈N ⊂ K be a Cauchy sequence. Therefore, for each ε > 0 there exists t0 ∈ N such that |xt − xs | < ε whenever s ≥ t ≥ t0 . In particular, |xt+1 − xt | → 0 whenever t → ∞. Conversely, suppose that |xt+1 − xt | → 0 as t → ∞ and write, xs − xt = xs − xs−1 + xs−1 − ... − xt+1 + xt+1 − xt . Combining all terms of the form xr − xr−1 for r = s, ...,t and using [Definition 1.3.1, (3)] it easily follows that |xs − xt | → 0 as t, s → ∞, and hence (xt )t∈N is a Cauchy sequence in K. In contrast with Proposition 2 above, in the next result, the completion of the field (K, | · |, d) is required. ∞

Proposition 3. Let (K, | · |, d) be a complete non-Archimedean field. The series verges to some S ∈ K if and only if xt → 0 as t → ∞.

∑ xt con-

t=0

4

Toka Diagana t

Proof. Let St =

∑ xr . It is then clear that xt = St − St−1. Thus, if St → S, then xt → 0 in K,

r=0

as t → ∞. Conversely, if xt → 0, then (St )t∈N is a Cauchy sequence, by Proposition 2. Now since K is complete, there exists S ∈ K such that St → S as t → ∞. Let (K, | · |) be a non-Archimedean valued field. Define the balls Ω(x, r), Ω0 (x, r), and the sphere S(x, r) of K by: Ω(x, r) := {y ∈ K : |x − y| ≤ r}, Ω0 (x, r) := {y ∈ K : |x − y| < r}, and S(x, r) := {y ∈ K : |x − y| = r}. In particular, we set Ωr = Ω(0, r), Ω0r = Ω(0, r), and Sr = S(0, r). Proposition 4. Let x ∈ K and let r > 0 be a real number. If z ∈ Ω(x, r), then Ω(z, r) = Ω(x, r). Proof. Let y ∈ Ω(z, r), that is, |z − y| ≤ r. Obviously, |y − x| = |(y − z) + z − x| ≤ max(|y − z|, |z − x|) ≤ r, and hence y ∈ Ω(x, r), or Ω(z, r) ⊂ Ω(x, r). Now, from the fact that z ∈ Ω(x, r) it is clear that x ∈ B(z, r). Arguing as previously, it follows that Ω(x, r) ⊂ Ω(z, r). From both inclusions it follows that Ω(x, r) = Ω(z, r). From Proposition 4, we obtain the following result: Theorem 1. Let x ∈ K and let r > 0 be a real number. The balls Ω(x, r), Ω0 (x, r) and the sphere S(x, r) are respectively closed and open in K. Note that subsets of K, which are both closed and open are commonly called clopens. In addition to the above, the unit balls Ω1 and Ω01 satisfy the following properties: Proposition 5. If (K, | · |) is a non-Archimedean field, then (1) Ω1 is a subring of K with identity 1K ; (2) K is the field of fractions of Ω1 ; (3) Ω01 is an ideal of Ω1 ; (4) Ω01 is the unique maximal ideal of Ω1 ; and (5) The quotient Ω1 /Ω01 is a field. The proof of Proposition 5 can be found in Gouvˆea [33]. Note that the balls Ω1 , Ω01 , and the quotient Ω1 /Ω01 are respectively called the valuation ring of (K, | · |), the valuation ideal of (K, | · |), and the residue field of K with respect to Ω01 .

Non-Archimedean Valued Fields

5

1.2.2 The t-Vector Space Kt The (K, | · |) be a non-Archimedean field. As for the n-dimensional real and complex numbers Rt and Ct , one defines Kt as follows: Kt := {x = (x1 , x2 , x3 , ...., xt ) : xr ∈ K, for r = 1, ...,t}. It is clear that Kt is a vector space over K. Moreover, one equips it with the absolute value |x|t = max |xr |, ∀x = (x1 , x2 , x3 , ...., xt ) ∈ Kt . 1≤r≤t

Clearly, for all x = (x1 , x2 , x3 , ...., xt ), y = (y1 , y2 , y3 , ...., yt ) ∈ Kt , |x + y|t = max |xr + yr | 1≤r≤t

≤ max max(|xr |, |yr |) 1≤r≤t

= max( max |xr |, max |yr |) 1≤r≤t

1≤r≤t

= max(|x|t , |y|t ), and hence | · |t is a non-Archimedean absolute value over Kt . In view of the above, it is easy to check that the following parallelogram law holds: for all x, y ∈ Kt , |x + y|t2 + |x − y|t2 ≤ 2 max(|x|t2 , |y|t2 ) with equality whenever |x|t 6= |y|t . It is not hard to see that many algebraic and topological properties of K can carried over to Kt . In particular, if (K, | · |) is complete, then so is (Kt , | · |t ). Let Ω(x, r) and S(x, r) denote respectively the ball and the sphere of Kt centered at x ∈ Kt with radius r > 0: Ω(x, r) := {y ∈ Kt : |x − y|t ≤ r}, and S(x, r) := {y ∈ Kt : |x − y|t = r}. Clearly, Ω(x, r) and S(x, r) are respectively clopens. Moreover, one can easily check that if x = (x1 , x2 , x3 , ...., xt ) ∈ Kt , then Ω(x, r) = Ω(x1 , r) × Ω(x2 , r) × Ω(x3 , r) × ... × Ω(xt , r). In addition to the above, an “inner product” is defined on Kt as follows: for all x = (x1 , x2 , x3 , ...., xt ), y = (y1 , y2 , y3 , ...., yt ) ∈ Kt , t

hx, yit :=

∑ xr yr = x1 y1 + x2y2 + ... + xt yt .

r=1

One can easily check that the “inner product” h·, ·it satisfies the Cauchy-Schwartz inequality, i.e.,

6

Toka Diagana |hx, yit | ≤ |x|t . |y|t , x, y ∈ Kt .

In contrast with the classical setting, the norm | · |t does not stem from the “inner product” h·, ·it . t

Each element x = (x1 , x2 , x3 , ...., xt ) in Kt can be written as x =

∑ xr er , where er =

s=1

(0, 0, 0, ..., 1 , ..., 0, 0, 0) with 1 at the rth term and zeros elsewhere. It is clear that such a | {z } decomposition is unique. The (finite) sequence (er )r=1,...,t will be called orthonormal base of Kt .

1.3 Construction of Q p 1.3.1 Introduction The p-adic numbers were discovered by Hensel at the end of the 19th century as a tool in number theory. Today, those numbers play a key role in many areas beyond number theory – among those areas are algebraic geometry, analysis, p-adic physics, p-adic quantum mechanics, representation theory, and many others. In what follows, we describe, by the means of completion arguments, the construction of Q p , the field of p-adic numbers. Nevertheless, one should point out that we will omit proofs of the classical results related to such a construction and refer the reader to the landmark book by Gouvˆea[33], which contains a comprehensive presentation on p-adic numbers and related issues. 1.3.2 The Field Q p Definition 3. Let p ∈ Z be a prime. A p-adic valuation on Z is a function v p : Z − {0} 7→ R, where v p (t) is the unique positive integer such that t = pv p (t) . q, where q is not divisible by p. Note that if v p is a p-adic valuation on Z, then it can be extended to the field of rational t numbers Q as follows: If x = ∈ Q − {0}, where t, s ∈ Z − {0}, one sets, v p (x) = v p (t) − s v p (s), and if x = 0, then one sets v p (0) = ∞. Definition 4. Let x ∈ Q. The p-adic absolute value of x is defined by ( p−v(x) if x 6= 0, |x| := 0 if x = 0, where v(x) is the valuation of x ∈ Q. It is not hard to check that (Q, | · |) is a non-Archimedean valued field. Moreover, from | · |, the p-adic absolute value, one defines the p-adic metric d on Q × Q by setting: d(x, y) = |x − y| for all x, y ∈ Q.

Non-Archimedean Valued Fields

7

Proposition 6. The non-Archimedean valued field (Q, | · |) is not complete, i.e., there exists at least one Cauchy sequence in (Q, | · |), which does not converge for | · |, the p-adic absolute value. Proof. See details of the proof in Gouvˆea [33]. Definition 5. The completion of (Q, | · |) is called the field of p-adic numbers and denoted by (Q p , | · |). The following characterization of p-adic numbers is due to Hensel4 : each x ∈ Q p can (uniquely) be expressed as x = ∑ at pt , t≥−t0

where 0 ≤ at ≤ p − 1. Moreover, v p (x) = −t0 , that is, |x| = pt0 . Example 1. In (Q p , | · |) (p ≥ 2 being a prime) consider the sequence xt = 1 + p2 + ... + pt . Clearly, |xt | = 1 for each t ∈ N. Moreover, 1 2 t lim (1 + p + ... + p ) − = 0. t→∞ 1 − p ∞

Set p = 2. In contrast with the classical case where

∑ 2t = ∞, in Q2 , such a series

t=0

converges to −1, that is, 1 + 4 + 8 + 16 + ... + .... = −1 in Q2 . Example 2. In (Q p , |·|) (p ≥ 2 being a prime) consider the sequence xt = 1+ p−2 +...+ p−t . Clearly, |xt | = pt for each t ∈ N. Moreover, lim 1 + p−2 + ... + p−t = ∞. t→∞

Definition 6. The valuation ring Ω1 = Ω(0, 1) of (Q p , | · |) is denoted by Z p and given by Z p := {x ∈ Q p : x = ∑ at pt , 0 ≤ at ≤ p − 1}. t≥0

Note that the unit ball Z p is also called ring of p-adic integers. We have the following characterization of Z p : Proposition 7. The set of p-adic integers Z p is a subring of Q p . Moreover, each element of Z p is the limit of a sequence of nonnegative integers and conversely, each Cauchy sequence in Q consisting of integers has a limit in Z p . Proof. See Baker [3]. 4

This characterization is also known as the Hensel decomposition.

8

Toka Diagana

1.3.3 Convergence of Power Series over Q p This subsection examines the convergence of power series in the field (Q p , | · |) of p-adic numbers. Consider the power series f (x) =

∑ αt xt ,

x ∈ Q p,

(1.3.1)

t∈N

where (αt )t∈N ⊂ Q p . First of all, consider the particular case when x = 1 in (1.3.1), that is, the series

∑ αt ,

αt ∈ Q p .

(1.3.2)

t∈N

Proposition 8. The series in (1.3.2) converges in Q p if and only if |αt | → 0 as t → ∞. Proof. This is a straightforward consequence of Proposition 3. More generally, we want find some (necessary) and sufficient conditions so that the series given in (1.3.1) converges, i.e., lim |αt | . |xt | = 0. t→∞

Definition 7. A number r is called the radius of convergence of the series in (1.3.1) if it converges whenever |x| ≤ r and diverges, otherwise, i.e., |x| > r. As in the classical context, define the radius of convergence of f as r :=

1 p . t lim |αt | p

Proposition 9. Let f (x) be the series given in (1.3.1). If r = 0, then f (x) converges at x = 0 only. If r = ∞, the series converges for each x ∈ Q p . Now if 0 < r < ∞ and lim |at |rt = 0, t→∞

then f (x) converges if and only if |x| ≤ r. Lastly, if 0 < r < ∞ and lim |at |rt 6= 0, then the t→∞

series f (x) converges if and only if |x| < r. Proof. See details in [33]. Example 3. Consider the power series

∑ xt

f (x) =

for each x ∈ Q p .

t∈N

One can easily check that r = 1 and lim |at |rt 6= 0, and hence f converges for each t→∞

x ∈ Q p such that |x| < 1, i.e., x ∈ Ω01 .

Example 4. Consider the power series (at = 1): f (x) =

xt

∑ pt

for each x ∈ Q p .

t∈N

One can easily see that r = 1/p and lim |at |rt = 1 6= 0, and hence the power series t→∞

converges if and only if |x| < 1/p, i.e., x ∈ Ω01/p .

Non-Archimedean Valued Fields

9

Other important power series are that of the p-adic logarithm and exponential. In contrast with the classical setting, the p-adic exponential is not defined and analytic everywhere in Q p . Definition 8. Let Ω0 (1, 1) = {x ∈ Z p : |x − 1| < 1} = 1 + pZ p . The p-adic logarithm is the power series defined, from Ω(1, 1) into Q p , by log p (x) :=

∑ (−1)t+1

t∈N

(1.3.3)

−p

Definition 9. Let Ω0 defined, from Ω0

(x − 1)t for each x ∈ Ω0 (1, 1). t

−p p p−1

= {x ∈ Z p : |x| < p p−1 }. The p-adic exponential is the power series

into Q p , by

−p

p p−1

exp p (x) :=

xt

∑ t!

for each x ∈ Ω0

−p

.

(1.3.4)

p p−1

t∈N

It can be checked that exp p (x + y) = exp p (x) exp p (y) whenever x + y ∈ Ω0

−p

.

p p−1

Moreover, exp p (x) belongs to 1 + pZ p , the domain of log p , with log p (exp p x) = x for each x ∈ Ω0

−p

.

p p−1

Similarly, log p (x + 1) belongs to Ω0

−p

with

p p−1

exp p (log p (x + 1)) = x + 1.

1.4 Construction of K((x)) Let K be any field (possibly R, C, or Q p ) and let K[x] be the ring of polynomials whose coefficients belong to K. Basically, each P ∈ K[x] can be expressed as: n

P(x) = ∑ at xt , t=0

where at ∈ K for t = 1, 2, ..., n. In that event, one can write P(x) = xv(P) . Q(x), where Q ∈ K[x] with Q(0) 6= 0. It is not hard to check that (1) v(P . Q) = v(P) + v(Q), for all P, Q ∈ K[x];

10

Toka Diagana

(2) v(P + Q) ≥ min(v(P), v(Q)), for all P, Q ∈ K[x]. Setting v(0) = +∞ and using (1)-(2) above, one can easily see that v is a valuation 5 over the ring K[x]. One can extend such a valuation over the field K(x) of rational fractions by P setting: For all f = ∈ K(x), Q v( f ) = v(P) − v(Q). Now for 0 < ς < 1, one sets, |P| = ςv(P) for each P ∈ K[x]. It is not hard to check that | · | is a non-Archimedean absolute value, which can be extended over K(x). The nonArchimedean field (K(x), | · |) is not complete; let (K((x)), | · |) denote its completion. The field (K((x)), | · |) is then called the field of formal Laurent series. Example 5. In Q p [x] (p ≥ 2 is a prime), let v(L) = −deg(L) if L is a polynomial and let ς = |p| = p−1 . The valuation v can be extended over Q p ((x)) as above. Consider f ∈ Q p ((x)) defined by f (x) =

L(x) , M(x)

where L(x) = a0 + a1 x + ... + an xn and M(x) = b0 + b1 x + ... + bm xm with ai , b j ∈ Q p for i = 0, 1, .., n, j = 0, 1, ..., m, and an , bm 6= 0. It is easy to check that | f | = p−v( f ) =

1 . pm−n

1.5 Bibliographical Notes This chapter introduces classical results on non-Archimedean valued fields. The uncovered topics related to non-Archimedean valued fields and related issues can be found in most of books devoted to non-Archimedean functional analysis, see, e.g., Gouvˆea[33], Koblitz[42], Mahler[50], Ochsenius & Schikhof[56], Robert[60], van Rooij[62], Vladimirov, Volovich & Zelenov[66], and many others. Basic tools of Subsection 1.3.3, that is, the convergence of power series were taken in both Gouvˆea[33] and Vladimirov, Volovich & Zelenov[66]. Similarly, basic and advanced tools on the algebra of analytic functions in the non-Archimedean setting can be found in those books. Section 1.4 related to the field of formal Laurent series was taken in Diarra[25].

5

Among possible valuations that can be defined over K[x], one may consider that defined through the degree of the polynomial, i.e., if P ∈ K[x], one defines v∞ (P) = −deg(P).

Chapter 2

Non-Archimedean Banach and Hilbert Spaces 2.1 Non-Archimedean Banach Spaces This section introduces non-Archimedean (free) Banach and Hilbert spaces. As for nonArchimedean valued fields, those Banach and Hilbert spaces play a crucial role in the composition of this book. For uncovered topics related to those non-Archimedean Banach spaces, we refer the reader to the excellent book by Rooij [62], which covers all topics related to those spaces. Moreover, for all questions on non-Archimedean Hilbert spaces, which are not treated here, we refer to the work of Diarra[24, 25]. Throughout the rest of this section, (K, k·k) denotes a complete non-Archimedean field. 2.1.1 Basic Definitions Definition 10. Let X be a vector space over K. A nonnegative real valued function k · k : X 7→ [0, ∞) is called a norm if: (1) kxk = 0 if and only if x = 0; (2) kλxk = |λ| kxk for all λ ∈ K and x ∈ X; and (3) kx + yk ≤ kxk + kyk for all x, y ∈ X. The norm k · k is called non-Archimedean if the statement (3) above can be replaced by the stronger condition (4) kx + yk ≤ max(kxk, kyk) for all x, y ∈ X. As for the absolute value of a non-Archimedean valued field, (4) above is called nonArchimedean triangle inequality. A normed vector space (X, k · k) satisfying [Definition 10, (1)-(2)-(4)] is called a non-Archimedean normed vector space1 . Moreover, as an immediate consequence of [Definition 10, (4)], the following holds: kx + yk = max(kxk, kyk) whenever kxk = 6 kyk. More generally, 1

A non-Archimedean normed vector space (X, k · k) is also called ultrametric normed vector space.

12

Toka Diagana

Proposition 10. Let (X, k · k) be a non-Archimedean normed vector space over K. If x1 , ...xt ∈ X such that kxs k 6= kxr k for all s 6= r, then kx1 + x2 + ... + xt k = max kxs k. 1≤s≤t

Proposition 11. Let (X, k · k) be a non-Archimedean normed vector space over K. Then2 kx + yk2 + kx − yk2 ≤ 2 max(kxk2 , kyk2 ) x, y ∈ X,

(2.1.1)

for all x, y ∈ X, with equality whenever kxk = 6 kyk. Definition 11. A non-Archimedean Banach space is a non-Archimedean normed vector space, which is complete.

2.1.2 Examples of Non-Archimedean Banach Spaces Example 6. Let (ωt )t∈I ⊂ R+ − {0} be a family of nonzero real numbers 3 . Define the K-vector space B∞ (I, K, ω) by B∞ (I, K, ω) := {x = (xt )t∈I ∈ KI : sup |xt |ωt < ∞}. t∈I

The space B∞ (I, K, ω) is equipped with the sup norm defined by kxk := sup |xt |ωt . t∈I

It is not hard to see that (B∞ (I, K, ω), k · k) is a non-Archimedean Banach space. Example 7. Let c0 (I, K, ω) ⊂ (B∞ (I, K, ω) be the subspace defined by c0 (I, K, ω) := {x = (xt )t∈I ∈ KI : lim |xt |ωt = 0}. t∈I

Clearly (c0 (I, K, ω), k · k) is a closed subspace of (B∞ (I, K, ω), k · k), and hence is also a non-Archimedean Banach space. Example 8. Let M be a compact (topological) space. Define the space of all continuous functions which go from M into K by C(M, K) := {u : M 7→ K, u is continuous}. We then equip C(M, K) with the sup norm: kuk∞ := sup |u(t)|. t∈M

Theorem 2. The normed vector space (C(M, K), k · k∞ ) defined above is a nonArchimedean Banach space. 2 3

Such an inequality is known as the non-Archimedean Parallelogram Law in a normed vector space. I is the index set. It is customary to take I = N. However the treatment of those non-Archimedean Banach space in the general case can be found in Diarra[24, 25].

Non-Archimedean Banach and Hilbert Spaces

13

2.2 Free Banach Spaces 2.2.1 Definitions Definition 12. A non-Archimedean Banach space (X, k · k) over K is said to be a free Banach space if there exists a family (et )t∈I of elements of X such that each element x ∈ X can be written in a unique fashion as a pointwise convergent series defined by x = ∑ xt et with lim xt et = 0, t∈I

t∈I

and kxk = sup |xt | ket k. The family (et )t∈I is then called an orthogonal base for X. If t∈I

ket k = 1, for all t ∈ I, then (et )t∈I is called an orthonormal base for X. 2.2.2 Examples Example 9. In Example 8, take M = Z p and K = (Q p , | · |) where p ≥ 2 is a prime number. Then the resulting space (C(Z p , Q p ), k · k∞ ) is a free Banach space. Indeed, consider the sequence of functions defined by: x(x − 1)(x − 2)(x − 3)....(x − t + 1) , t ≥ 1, t! f0 (x) = 1. ft (x) =

It is well-known [46] that the family ( ft )t∈N is an orthonormal base, i.e., k ft k∞ = 1. Moreover, for each u ∈ C(Z p , Q p ) has a unique uniformly convergent decomposition defined by ∞

u(x) = ∑ ct ft (x), ct ∈ Q p t=0

with |ct | 7→ 0 as t 7→ ∞ and kuk∞ = sup |ct |. t∈N

Let (X, k · k) be a free Banach space over K. If X∗ denotes the (topological) dual of E, then the following spaces are isomorphic: X ' c0 (I, K, (ket kt∈I )), and X∗ ' B∞ (I, K, (

1 )t∈I ). ket k

Example 10. The space c0 (I, K, ω) given in Example 7 is a free Banach space (see details in the next subsection). Example 11. Let (K, | · |) be a complete non-Archimedean field. Then the t-vector space Kt previously defined is a free Banach.

14

Toka Diagana

Remark 1. Let (et )t∈I be an orthogonal base for the free Banach space (X, k · k). Define et0 ∈ X∗ (topological dual of X) by: x = ∑ xt et , et0 (x) = xt . t∈I

1 . Furthermore, each x0 ∈ X∗ can be expressed as a pointwise ket k convergent series defined by x0 = ∑hx0 , et iet0 , and It turns out that ket0 k =

t∈I

kx0 k = sup t∈I

|hx0 , et i| . ket k

2.3 Non-Archimedean Hilbert Spaces 2.3.1 Introduction This section is devoted to the so-called non-Archimedean Hilbert spaces Eω . They play a key role throughout the book. Among other things, we will see that the norm k · k of a non-Archimedean Hilbert space Eω does not stem from its corresponding inner product h·, ·i. Moreover, it may happen that |hx, xi| < kxk2 for some x ∈ Eω . In addition to that Eω contains isotropic vectors x 6= 0, that is, hx, xi = 0. For more on these spaces and related issues we refer the reader to [4], [24], and [25]. 2.3.2 Non-Archimedean Hilbert Spaces Let ω = (ωt )t∈N ⊂ K be a sequence of non-zero elements and define the space Eω = c0 (N, K, ω), that is, Eω := {x = (xt )t∈N , ∀t, xt ∈ K and lim |xt ||ωt |1/2 = 0}. t→∞

It can be easily shown that x = (xt )t∈N ∈ Eω if and only if lim x2 ωt t→∞ t

= 0.

Moreover, Eω is a non-Archimedean Banach space over K when it is endowed with the norm x = (xt )t∈N ∈ Eω , kxk = sup |xt ||ωt |1/2 .

(2.3.1)

t∈N

It is also clear that Eω is a free Banach space and has a canonical orthogonal base, namely, (et )t∈N , where et is the sequence whose terms are 0 except the t-th term, which is 1, in other words, et = (δts )s∈N , where δts is the Kronecker symbol. We shall make

Non-Archimedean Banach and Hilbert Spaces

15

extensive use of this canonical orthogonal base throughout the rest of this book. For each t, ket k = |ωt |1/2 . If |ωt | = 1, we shall refer to (et )t∈N as the canonical orthonormal base of Eω . Let h·, ·i : Eω × Eω → K be the linear form defined for all u, v ∈ Eω , u = (ut )t∈N , v = (vt )t∈N , by hu, vi :=

∑ ωt ut vt .

(2.3.2)

t∈N

Clearly, h·, ·i is a symmetric, bilinear, and non-degenerate linear form on Eω and satisfies the Cauchy-Schwarz’s inequality: ∀u, v ∈ Eω , |hu, vi| ≤ kuk . kvk.

(2.3.3)

Remark 2. The K-form h·, ·i defined in (2.3.2) is called non-Archimedean inner product. On the canonical orthogonal base, the following holds:    0 if t 6= s het , es i = ωt δts =   ω if t = s. t Definition 13. The space (Eω , k·k, h·, ·i), where k·k, h·, ·i are respectively the norm defined in (2.3.1) and the inner product in (2.3.2), is called a non-Archimedean Hilbert space. Proposition 12. Let ω = (ωt )t∈N ⊂ K be a sequence of nonzero terms and let Eω be the corresponding non-Archimedean space. If (et )t∈N denotes the canonical orthogonal base of Eω , then, for all x ∈ Eω , ∞ (< x, e >)2 t 2 (2.3.4) ≤ kxk . ∑ t=0 ωt Proof. Observe that

ω2 x2 (< x, et >)2 = t t = ωt xt2 . ωt ωt

Moreover, lim |ωt xt2 | = 0, by x = ∞

t→∞

(< x, et >)2 |ωt |1/2 . |xt | = 0). Hence, the series ∑ converges. ∑ xt et ∈ Eω (lim t→∞ ωt t∈N t=0 Now ∞ (< x, e >)2 ∞ t = ∑ ωt xt2 ∑ t=0 t=0 ωt ≤ sup |ωt ||xt |2 t≥0

= sup(|ωt |1/2 |xt |)2 t≥0

= ( sup |ωt |1/2 |xt | )2 t≥0

= kxk2 .

16

Toka Diagana Consequently,

Corollary 1. If ω = (ωt )t∈N ⊂ K where ωt = 1K for each t ∈ N and if Eω is the corresponding non-Archimedean space, then, for all x ∈ Eω , ∞ ∑ (< x, et >)2 ≤ kxk2 . t=0 Another example of non-Archimedean Hilbert space consists of considering C(Z p , Q p ) as in Example 9 and put the inner product (symmetric, bilinear, non degenerate form) is ∞

∞

t=0

t=0

defined as follows: for all u = ∑ ut ft , and v = ∑ vt ft in C(Z, Q p ), ∞

hu, vi := ∑ ut vt . t=0

It is clear that the inner product h·, ·i given above is well-defined as both ut and vt 7→ 0 in Q p as t → ∞, and the Cauchy-Schwartz inequality holds |hu, vi| ≤ kuk∞ . kvk∞ . In view of the above, it easily follows that h ft , fs i = δts where δts are the classical Kronecker symbols. The triplet (C(Z p , Q p ), k · k∞ , h·, ·i) will be called a p-adic or non-archimedean Hilbert space. One must point out that the norm on C(Z p , Q p ) does not stem from its corresponding inner product. 2.3.3 The Hilbert Space Eω1 × Eω2 × ... × Eωt Let (K, | · |) be a complete non-Archimedean field with characteristic char(K) zero. Examples of such fields include Q p (p ≥ 2 being prime) the field of p-adic numbers. Let ωτ = (ωτs )s∈N with τ = 1, 2, ...,t be (t being fixed) t sequences of nonzero terms in K. For each τ (τ = 1, 2, ...t) we consider the corresponding non-Archimedean Hilbert space Eωτ equipped with its usual topologies (see (2.3.1)-(2.3.2)). Let Et denote the direct sum Eω1 × Eω2 × ... × Eωt of Eω1 , Eω2 , ..., and Eωt , respectively. We now equip the vector space Et with the non-Archimedean norm defined by ||(u1 , ..., ut )||n := max(ku1 k, ..., kut k),

(2.3.5)

for all (u1 , ..., ut ) ∈ Et . Arguing that each (Eωτ , k.k) for τ = 1, 2, ...t is a non-Archimedean Banach space it follows that (Et , || · ||t ) is also a non-Archimedean Banach space. It also clear that Et (t fixed) is a free Banach space with orthogonal base: {(Fsτ )s∈N , τ = 1, 2, ...,t} where Fsτ = (0, 0, ...eτs , ..0, 0, 0) whose terms are 0 except the s-th term, which is eτs ((eτs )s∈N being the | {z }

Non-Archimedean Banach and Hilbert Spaces

17

canonical orthogonal base of Eωτ for τ = 1, 2, ..,t). One can easily check that kFsτ kt = keτs k = |ωτs |1/2 for all s ∈ N and for each τ = 1, 2, ...,t. We confer to {(Fsτ )s∈N , τ = 1, 2, ...,t} as the canonical orthogonal base for Et . Let h·, ·it : Et × Et 7→ K be the K-bilinear form defined by t

h(u1 , ..., ut ), (v1 , ..., vt )it :=

∑ huτ , vτ i,

(2.3.6)

τ=1

for all (u1 , ..., ut ), (v1 , ..., vt ) ∈ Et where h·, ·i is the inner product of each Eωτ for τ = 1, 2, ...,t. Then, h·, ·it is a symmetric, non-degenerate form on Et × Et . Definition 14. The space Et := Eω1 × Eω2 × ... × Eωt endowed with the norm given in (2.3.5) and the K-bilinear form h·, ·it given in (2.3.6) is called a non-Archimedean Hilbert space. Proposition 13. For every (u1 , ..., ut ), (v1 , ..., vt ) ∈ Et , the Cauchy-Schwarz inequality holds, that is, |h(u1 , ..., ut ), (v1 , ..., vt )it | ≤ ||(u1 , ..., ut )||n . ||(v1 , ..., vt )||t . Proof. Suppose t = 2. For all (u, v), (x, y) ∈ Eω1 × Eω2 , |h(u, v), (x, y)i2 | = |hu, xi + hv, yi| ≤ max(|hu, xi|, |hv, yi|) ≤ max(kuk kxk, kvk kyk) ≤ max(||(u, v)||2 kxk, ||(u, v)||2 kyk) ≤ max(||(u, v)||2 ||(x, y)||2 , ||(u, v)||2 ||(x, y)||2 ) = ||(u, v)||2 ||(x, y)||2 . When t ≥ 3, one follows the same lines as above. If M ⊂ Eω × Eω is a subspace, then its orthogonal complement M ⊥ with respect the K-bilinear form h·, ·i2 is defined by M ⊥ := {(u, v) ∈ Eω × Eω : h(u, v), (x, y)i2 = 0, ∀(x, y) ∈ M}. One can easily check that M ⊥ is closed. This is actually a consequence of the continuity of the bilinear form defined by Φ(x,y) : Eω × Eω 7→ K and Φ(x,y) (u, v) = h(u, v), (x, y)i2 , ∀(u, v) ∈ Eω × Eω , where (x, y) ∈ Eω × Eω is fixed.

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Toka Diagana

2.4 Bibliographical Notes This chapter is entirely devoted to non-Archimedean Banach spaces, free Banach spaces, and non-Archimedean Hilbert spaces. Algebraic properties of those spaces can be found in Ochsenius & Schikhof [56], and van Rooij [62]. For more on free Banach spaces and non-Archimedean Hilbert spaces, we refer the reader to Diagana [17], especially Diarra [24, 25], and Khrennikov [37]. Details on the Banach space C(Z p , Q p ) can be found in Kochubei [46]. Subsection 2.3.3 was taken in Diagana [17].

Chapter 3

Non-Archimedean Bounded Linear Operators 3.1 Introduction This chapter studies the class of bounded linear operators on non-Archimedean (free) Banach and Hilbert spaces. Section 3.2 examines bounded linear operators on an arbitrary non-Archimedean Banach space while Section 3.3 considers those operators on Eω , in particular, the existence of an adjoint operator for a given bounded linear operator is screened. Section 3.4 presents some recent results on perturbation of bases of Eω obtained by Diagana & Ramaroson in [12]. Section 3.5 is concerned with non-Archimedean analogues of the classical Hilbert-Schmidt operators as well as their properties (Diagana et al. [4]). Some of the results go along the classical line and others deviate from it. For the most part, the statements of the results are inspired by their classical counterparts, however, their proofs may depend heavily on the non-Archimedean nature of Eω and the ground field K. Among other things, the definition of a Hilbert-Schmidt operator in the non-archimedean context depends on the base; further, Hilbert-Schmidt operator has an adjoint, which is again a Hilbert-Schmidt operator and that both have the same Hilbert-Schmidt norm. B2 (Eω ), the collection of all Hilbert-Schmidt operators on Eω is a two-sided ideal in the ring B0 (Eω ) of all bounded operators, which have adjoint. In addition to that it will be shown that every Hilbert-Schmidt operator is completely continuous in the sense that it is a limit, in B(Eω ), of a sequence of operators of finite ranks. As in the classical setting, a natural inner product is considered for Hilbert-Schmidt operators and we show that it satisfies the Cauchy-Schwarz inequality. We define the trace of an operator. As in the classical context, the trace may or may not exist. It is then shown that a Hilbert-Schmidt operator has a trace. We next illustrate those abstract results with several examples at the end of Section 3.5.

3.2 Bounded Linear Operators on Non-Archimedean Banach Spaces Throughout this section, unless otherwise, (K, | · |) denotes a complete non-Archimedean field.

20

Toka Diagana

3.2.1 Basic Definitions Let (X, k · kX ) and (Y, k · kY ) be non-Archimedean Banach spaces over K, respectively. A linear operator A : (X, k · kX ) 7→ (Y, k · kY ) is a transformation, which maps linearly X into Y. Definition 15. A linear operator A : X 7→ Y is said to be bounded if there exists K ≥ 0 such that kAukY ≤ K . kukX for each u ∈ X. The collection of bounded linear operators from X into Y is denoted B(X, Y). When X = Y, this is simply denoted B(X). Remark 3. If A : X 7→ Y is a bounded linear operator, then its norm kAk is defined by kAukY kAk := sup . (3.2.1) kukX u6=0 Clearly, 0 ∈ B(X, Y) with k0k = 0. Furthermore, if A, B ∈ B(X, Y) and if λ ∈ K, then A + B, λB, AB belong to B(X, Y), and the following hold: (1) kA + Bk ≤ kAk + kBk; (2) kλBk = |λ| . kBk; (3) kABk ≤ kAk . kBk. Consequently, (B(X, Y), k · k) is a normed vector space. Lemma 1. The space (B(X, Y), k · k) of bounded linear operators is non-Archimedean normed vector space. Proof. Let A, B ∈ (B(X, Y), k · k). We want to show that the operator norm k · k satisfies the non-Archimedean inequality. Indeed, kAu + BukY kA + Bk = sup kukX u6=0 kAukY kBukY , ≤ sup max kukX kukX u6=0 ! kAukY kAukY , sup = max sup u6=0 kukX u6=0 kukX = max (kAk, kBk) . Theorem 3. If (X, k.kX ), (Y, k.kY ) are non-Archimedean Banach space over K, then (B(X, Y), || · ||) is a non-Archimedean Banach space. The proof of Theorem 3 is similar to that of the classical setting. However, for the sake of clarity, we will provide the reader with it.

Non-Archimedean Bounded Linear Operators

21

Proof. In view of the previous facts and Lemma 1, it remains to prove that (B(X, Y), k · k) is complete. Let (At )t∈N ∈ B(X, Y) be a Cauchy sequence. Therefore, for each ∀ε > 0 there exists t0 (ε) ∈ N such that kAs − At k ≤ ε whenever s,t ≥ t0 (ε). Consequently, for each x ∈ X, (At x)t∈N is a Cauchy sequence in Y. Since Y is complete, (At x)t∈N converges strongly in Y as t → ∞. Define A : X 7→ Y by setting Ax := lim At x in Y. t→∞

(1) Linearity of A: If λ, µ ∈ K, ∀t ∈ N, then At (λx + µy) = λAt x + µAt y, for all x, y ∈ X. Thus, when t goes to ∞, it follows that A(λx + µy) = λAx + µAy, for all x, y ∈ X. (2) Boundedness of A: Fist of all, note that kAxkY = lim kAt xkY . Next, |kAt k − kAs k| ≤ t→∞

kAt − As k ≤ ε whenever s ≥ t ≥ t0 (ε), and hence (kAt k)t∈N is a Cauchy sequence in R. Clearly, (kAt k)t∈N converges, and hence is bounded, i.e., there exists M ≥ 0 such that kAt k ≤ M for each t ∈ N. Now kAt xkY ≤ M . kxk for all t ∈ N and x ∈ X, and hence kAxkY ≤ M . kxk for all x ∈ X, consequently, A is bounded. To complete the proof we have to show that kAt − Ak 7→ 0 as t → ∞. In view of the above, for each x ∈ X, kAt x − AsxkY ≤ kAt − Ask kxkX , and hence kAt x − As xkY ≤ εkxkX

(3.2.2)

whenever s ≥ t ≥ t0 (ε). Letting s → ∞ in (3.2.2) it follows that kAt x − AxkY ≤ εkxkX whenever n ≥ t0 (ε), that is, kAt − Ak → 0 as t → ∞. 3.2.2 Examples Example 12. Let K = Q p equipped with the p-adic absolute value and suppose that X = Y = C(Z p , Q p ), is the non-Archimedean Banach space appearing in Example 9 equipped with its corresponding sup norm. Define Mγ : C(Z p , Q p ) 7→ C(Z p , Q p ), Mγ φ := γ(x)φ,

(3.2.3)

where γ ∈ C(Z p , Q p ). In view of the above, kMγ φk∞ ≤ K . kφk∞ , for each φ ∈ C(Z p , Q p ), where K = kγk∞ . Consequently, Mγ is a bounded linear operator on C(Z p , Q p ). Note that Mγ is commonly called multiplication operator with potential γ.

22

Toka Diagana

Example 13. Let (K, |·|) be an arbitrary (complete) non-Archimedean field and let X = Y = Kt , equipped with its natural topology (see Chapter 1). Let M = (ars )r,s=1,..,t be the t × t square matrix, i.e., t

Mer =

∑ asr es .

s=1

One can easily see that M is a bounded linear operator with norm kMk = max |ars |. 1≤r,s≤t

3.2.3 The Banach Algebra B(X) A vector space (Y, +) over K is called an algebra if for each x, y ∈ Y, a unique product x ∗ y ∈ Y can be defined such that (1) (2) (3) (4)

(x ∗ y) ∗ z = x ∗ (y ∗ z); x ∗ (y + z) = x ∗ y + x ∗ z; (x + y) ∗ z = x ∗ z + y ∗ z; λ(x ∗ y) = (λx) ∗ y = x ∗ (λy);

for all x, y, z ∈ Y and λ ∈ K. An algebra (Y, +, ∗) with norm k · k such that kx ∗ yk ≤ kxk . kyk for all x, y ∈ Y, is called a normed algebra. A normed algebra (Y, +, ∗, k · k) is called a Banach algebra if it is complete for k · k. Moreover, a Banach algebra (Y, +, ∗, k · k) is called a Banach algebra with unity whether it contains an element e with kek = 1 such that e ∗ x = x ∗ e = x for each x ∈ Y. In view of the above, it is clear that (B(X), +, ◦, k · k), the collection of all bounded linear operators on X equipped with ◦, the composition operator, is a Banach algebra with unity. 3.2.4 Further Properties of Bounded Linear Operators Let (X, k · kX ), (Y, k · kY ) be non-Archimedean Banach spaces over K. If A ∈ B(X, Y), then its kernel N(A) and range R(A) are respectively defined by N(A) = {x ∈ X : Ax = 0}, and R(A) = {Ax ∈ Y : x ∈ X}. Note that both N(A) ⊂ X and R(A) ⊂ Y are (vector) subspaces. Furthermore, R(A) may or may not be closed while N(A) is always closed. Definition 16. If A ∈ B(X, Y), then (1) A is said to be one-to-one if N(A) = {0}; (2) A is said to be onto if R(A) = Y;

Non-Archimedean Bounded Linear Operators

23

(3) A is said to be invertible if it is both one-to-one and onto. If A is invertible, then there exists a unique bounded linear operator denoted A−1 : Y 7→ X called the inverse of A such that A−1 A = IX , and AA−1 = IY , where IX and IY are the identity operators on X and Y, respectively. When X = Y, the identity operator is denoted by I. Lemma 2. Let (X, k · kX ) is a non-Archimedean Banach space over K and let A ∈ B(X). If kAk < 1, then I − A is invertible, and (I − A)−1 = ∑ At

(3.2.4)

t≥0

with kI − Ak ≤ 1. Proof. First of all, let us check that the series

∑ At is well defined. This is actually equiva-

t≥0

lent to the convergence of At to zero in B(X), as t → ∞. Clearly, kAt k ≤ kAt−1 kkAk ≤ kAkt for each t ∈ N, and hence kAkt → 0 as t → ∞, by kAk < 1. In view of the above, the series ∑ At is absolutely convergent. Denote its sum by S := ∑ At . Clearly, S ∈ B(X). Indeed,

t≥0

t≥0

t

kSk = k lim ∑ As k t→∞

s=0 t

≤ lim k ∑ As k t→∞

s=0

≤ lim max(1, kAk, kA2 k, ...., kAt k) t→∞

≤ lim max(1, kAk, ...., kAkt ) t→∞

= 1. Moreover, t

(I − A)S = (I − A) lim ∑ As = lim (I − At+1) = I. t→∞

s=0

t→∞

Similarly, it can be easily shown that R(I − A) = X, and hence S = (I − A)−1 ∈ B(X) with k(I − A)−1k ≤ 1. Let (X, k · k) be a non-Archimedean Banach space over K and let A ∈ B(X). The resolvent set of A is the set of all λ ∈ K such that the operator A − λI has an inverse operator (A − λI)−1 ∈ B(X). The spectrum σ(A) of A is the set of all λ ∈ K such that A − λI is not invertible, that is, σ(A) = K − ρ(A).

24

Toka Diagana

Example 14. Let X = C(Z p , Q p ) equipped with the sup norm. Consider the multiplication operator given in Example 13 (with γ(x) = x) defined by M : C(Z p , Q p ) 7→ C(Z p , Q p ), Mφ(x) := x . φ(x). It is not hard to check that both the spectrum and the resolvent set of M are respectively given by σ(M) = Z p , and ρ(M) = Q p − Z p. A scalar λ ∈ K is called eigenvalue of an operator A ∈ B(X) if there exists 0 6= u ∈ X such that Au = λu. In this event, u is called an eigenvector associated with the eigenvalue λ. The set of all eigenvalues is called the point spectrum and denoted by σP (A). As in the classical setting, the following holds: Proposition 14. Let A ∈ B(X). Then σP (A) ⊂ σ(A). Proof. Let λ ∈ σP (A) and let 0 6= u ∈ X be an eigenvector associated with λ. Clearly, A − λI is not one-to-one, by (A − λI)u = 0 and (A − λI)0 = 0 with u 6= 0, and hence λ ∈ σ(A). Example 15. Let p be an odd prime. Let us equip Q p × Q p with the non-Archimedean norm given by k(x, y)k = max(|x|, |y|) for all x, y ∈ Q p . Consider the 2 × 2 square matrix defined by ! ab T= , a, b ∈ Q p − {0}. ba One can easily check that: (2) σ(T ) = σP (T ) = {a − b, a + b}; (3) ρ(T ) = Q p − {a − b, a + b}; (3) kT k = max(|a − b|, |a + b|) = max(|a|, |b|). We will retrieve the matrix T in Chapter 6 through a study of its functions. Let A ∈ B(X). Define the resolvent RAλ for each λ ∈ ρ(A) by RAλ = (A − λI)−1. By definition of an element λ ∈ ρ(A), RAλ = (A − λI)−1 ∈ B(X). As in the classical setting, the so-called resolvent equation holds in the nonArchimedean context: Proposition 15. Let A ∈ B(X). Then the resolvent equation holds, that is, RAλ − RAµ = (λ − µ) . RAλ RAµ for all λ, µ ∈ ρ(A). Proof. Write RAλ − RAµ = RAλ [(A − µI) − (A − λI)]RAµ = (λ − µ)RAλ RAµ .

Non-Archimedean Bounded Linear Operators

25

3.3 Bounded Linear Operators on Hilbert Spaces Eω 3.3.1 Introduction This section examines bounded linear operators on Eω . Our first task consist of studying the decomposition of those operators as (non-Archimedean) infinite matrices. The second task, consists of taking a closer look into the crucial issue of the existence or not of the adjoint with respect to the non-Archimedean inner product given in (2.3.2). 3.3.2 Representation of Bounded Operators By Infinite Matrices This subsection is mainly concerned with the decomposition of bounded linear operators in terms of infinite matrices. For a Banach space (E, k·k), let (E∗ , k·k∗ ) be its (topological) dual. For (u, v) ∈ E×E∗ , define the linear operator (v ⊗ u) by ∀x ∈ E, (v ⊗ u)(x) := v(x)u = hv, xiu. More generally, let E and F be free Banach spaces over the same field K with canonical orthogonal bases (et )t∈I and ( ft )t∈J , respectively (I, J being arbitrary index sets). Now if f 0 ∈ E∗ (dual of E), then one defines the linear functional f 0 ⊗ g : E 7→ F by setting: ( f 0 ⊗ g)(h) := h f 0 , hig withk f 0 ⊗ gk = k f 0 kkgk. In particular if (et0 )t∈I is the dual canonical orthogonal base for E∗ it can be easily seen that (et0 ⊗ fs )(t,s)∈I×J ∈ B(E, F), moreover ket0 ⊗ fsk =

k fs k . ket k

Clearly, if A ∈ B(E, F), then it can be decomposed through the base ( ft )t∈J as follows: For all t ∈ I, Aet = ∑ ast fs with lim |ast |k fs k = 0, ∀t ∈ I. s∈J

s∈J

Consequently, A = ∑ ats et0 ⊗ fs . Furthermore, from the definition of the norm of A, i.e., ts

kAxk it follows that kAk := sup x6=0 kxk kAet k ket k t |ast |k fs k . = sup sup ket k t s

kAk = sup

The previous discussion can be formulated by:

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Toka Diagana

Proposition 16. If A ∈ B(E, F), then it can be written in a unique fashion as a pointwise convergent series A = ∑ ats et0 ⊗ fs , t ∈ I, lim |ats |k fs k = 0. Moreover, kAk = sup sup t

s

(3.3.1)

s

ts

|ast |k fs k . ket k

Example 16. Suppose that E = F and let A = ∑ats et0 ⊗ es , where ats = λ ∈ K − {0} for all ts

(t, s) ∈ I × J. If ket k = α > 0 for all t, then A is a bounded linear operator. Moreover, kAk = |λ|. If (E, k · k) is a free Banach, then for every linear operator A on E, the domain D(A) of A is defined by D(A) := {x = (xt ) ∈ E : lim |xt |kAet k = 0}.

(3.3.2)

t

In fact, if A is a bounded linear operator, then D(A) = E. Indeed, for each x = ∑xt et in t

E, |xt |kAet k ≤ kAk|xt |ket k = kAk|xt ||ωt k1/2 . Note that there are linear operators on E such that D(A) 6= E. Those operators are called unbounded linear operators and will be studied in Chapter 3. Throughout the next subsection, we suppose that E = F = Eω , I = J = N, and study the existence of the adjoint of a bounded linear operator on Eω . 3.3.3 Existence of the Adjoint In contrast with the classical operator theory, we will see that there exist bounded linear operators on Eω , which do not have adjoint with respect to the non-Archimedean inner product given in (2.3.2). Therefore, a necessary and sufficient condition, which ensures the existence of the adjoint will be given. For more on the present setting we refer the reader to [4], [24], and [25]. Let ω = (ωt )t∈N ⊂ K be a sequence of nonzero elements and let Eω be its corresponding non-Archimedean Hilbert space. Let A, B be bounded linear operators on Eω . As we have previously seen, both A and B can be decomposed as follows: A=

∑

ats e0s ⊗ et , ∀s ∈ N, lim |ats ||ωt |1/2 = 0,

∑

bts e0s ⊗ et , ∀s ∈ N, lim |bts ||ωt |1/2 = 0.

t,s∈N

and B=

t,s∈N

t→∞

t→∞

Non-Archimedean Bounded Linear Operators

27

Definition 17. The linear operator B given above is called the adjoint of A if and only if, hAφ, ψi = hφ, Bψi, for all φ, ψ ∈ Eω , where h·, ·i is the inner product of Eω given in (2.3.2). Remark 4. (1) If the adjoint of an operator exists, then it is unique. The uniqueness of the adjoint is actually guaranteed by the fact that the inner product h·, ·i in (2.3.2) is non-degenerate. (2) It is sufficient to define the adjoint of A using the canonical basis (et )t∈N for Eω . Thus B is an adjoint for A if and only if: hAet , es i = het , Bes i, ∀t, s ∈ N.

(3.3.3)

(3) The equation (3.3.3) is equivalent to: bts = wt−1ws ast for all t, s ∈ N. Moreover, since lim |bts ||ωt |1/2 = 0, ∀s ∈ N, the operator A has an adjoint if and only if t→∞

lim

s→∞

|ats | |ωs |1/2

= 0, ∀t ∈ N.

(3.3.4)

(4) The adjoint of an operator A is denoted by A∗ . Theorem 4. Let A be the bounded linear operator given above. Then A has an adjoint A∗ ∈ B(Eω ) if and only (3.3.4) holds. Under (3.3.4), the adjoint A∗ of A can be uniquely expressed by A∗ =

∑

wt−1 ωs ast e0s ⊗ et .

(3.3.5)

(t,s)∈N×N

Proof. Write A∗ =

et0 ⊗ es , then A∗ is the adjoint of A if and only if hAes , et i =

∑ ∑ bst

t∈N s∈N

hes , A∗ et i. In other terms: + *

∑ aks ek , et

= ats ωt =

k∈N

*

es , ∑ bkt ek

+

= bst ωs , ∀s,t ∈ N.

k∈N

This is equivalent to bst = ω−1 s ωt ats for all s,t ∈ N. Moreover, for all t, lim |bst | |ωs |1/2 = 0.

s→∞

This is equivalent to lim

t→∞

|ast | |ωt |1/2

= 0, for all s ∈ N.

The collection of all bounded linear operators on Eω whose adjoint do exist with respect to the non-Archimedean inner product h·, ·i given in (2.3.2), is denoted by B0 (Eω ). Namely, B0 (Eω ) := {

∑

t,s∈N

ats e0s ⊗ et ∈ B(Eω ) : lim

|ats |

s→∞ |ωs |1/2

= 0, ∀t ∈ N}.

28

Toka Diagana

Remark 5. Note that B0 (Eω ) is stable under the operation of taking an adjoint. Namely, for all A, B ∈ B0 (Eω ) and λ ∈ K, the following hold: (1) (A + B)∗ = A∗ + B∗ ; (2) (AB)∗ = B∗ A∗ ; (3) (λA)∗ = λA∗ ; (4) (A∗ )∗ = A; (5) kAk = kA∗ k. 3.3.4 Examples of Bounded Operators with no Adjoint Example 17. Let K = (Q p , |·|). Suppose that ωt = 1+ p1+t for each t ∈ N. Define the linear operator A : Eω 7→ Eω by: A = ∑ ats (e0s ⊗ et ), t,s

where ats = p1+t for all t, s ∈ N. Proposition 17. The operator A defined above is bounded and does not have an adjoint. Proof. The operator A = ∑ p1+t (e0s ⊗ et ) is well-defined. Indeed, ∀s, lim |ats | . |ωt |1/2 = t→∞

t,s

lim p−(1+t) = 0. Furthermore,

t→∞

1 |ats | . |ωt |1/2 = sup p−(1+t) = < ∞. 1/2 p |ωs | t,s∈N t,s∈N

kAk = sup

|ats | 6= 0. Indeed, s→∞ |ωs |1/2

It remains to prove that ∀t, lim

∀t, lim

|ats |

s→∞ |ωs |1/2

= lim p−(1+t) = p−(1+t) 6= 0, s→∞

hence A does not have an adjoint. Example 17 is a particular case of the next example. Example 18. Let (K, |·|) be a non-Archimedean valued field. Suppose that |ωt | = 1 for each t ∈ N. Define the linear operator A : Eω 7→ Eω by: A = ∑ ats (e0s ⊗ et ), where ats = ϑt for t,s

all t, s ∈ N with |ϑt | 6= 0 for each t ∈ N and lim |ϑt | = 0. t→∞

Proposition 18. The operator A defined above is bounded and does not have an adjoint. Proof. The proof is similar to that of Proposition 17, and therefore left to the reader.

Non-Archimedean Bounded Linear Operators

29

3.4 Perturbation of Bases Let an orthogonal base (hs )s∈N be given and consider a sequence of vectors ( fs )s∈N , not necessarily an orthogonal base, in Eω such that the difference fs − hs is small in a certain sense. We study some sufficient conditions under which the family ( fs )s∈N is a base of Eω . Theorem 5. Let (hs )s∈N be an orthogonal base and let ( fs )s∈N be a sequence of vectors in Eω satisfying the following condition: There exists α ∈ (0, 1) such that for every x = ∑ xs hs ∈ Eω, s∈N

sup |xs | k fs − hs k ≤ α . sup |xs | khs k = α . kxk . s∈N

s∈N

Then ( fs )s∈N is an orthogonal base. Moreover, for any y =

∑ ys fs ∈ Eω,

kyk =

s∈N

sup (|ys | . k fs k) . s∈N

Proof. We first observe that, if x = hs , then, the condition implies that for each s, k fs − hs k ≤ α khs k < khs k, hence k fs k = khs k. Next, for any x = ∑ xs hs ∈ Eω , |xs | k fs k = |xs | khs k, that is, lim (|xs | . k fs k) = 0. Theres→∞

s∈N

fore, the operator defined by Ax =

∑ xs fs

s∈N

is well-defined and satisfies Ahs = fs . Moreover

kx − Axk = ∑ xs (hs − fs )

s∈N

≤ sup (|xs | . khs − fs k) s∈N

≤ α . sup (|xs | . khs k) s∈N

= α . kxk . It follows that kI − Ak ≤ α < 1, and hence A is invertible, by Theorem 2. It remains to show that A is isometric. In view of the above, observe that the inequalities: kx − Axk ≤ α kxk < kxk imply that kAxk = kxk for any x ∈ Eω , hence, A is isometric. Consequently, ( fs )s∈N is an orthogonal base. Moreover, for any y = ∑ ys fs ∈ Eω s∈N

kyk = A( ∑ ys hs ) = ∑ ys hs = sup (|ys | . khs k).

s∈N

s∈N

s∈N Among other things, the following is an immediate consequences of Theorem 5. Corollary 2. Let (es )s∈N be the canonical base for Eω and let ( fs )s∈N ⊂ Eω be a family of vectors such that

30

Toka Diagana ket − ft k sup < 1. ket k t∈N

(3.4.1)

Then ( fs )s∈N is also an orthogonal base for Eω .

Theorem 6. Let (hs )s∈N be an orthogonal base, C ∈ B(Eω ) invertible such that C−1 = kCk−1 . Suppose that ( fs )s∈N is a sequence of vectors in Eω satisfying the following condition k fs −Chs k < kCk , sup khs k s∈N then ( fs )s∈N is an orthogonal base. Proof. We first observe that for any s, k fs k ≤ kCk khs k . For any x =

∑ xs hs ∈ Eω :

s∈N

k fs k s→∞ khs k ≤ kCk . lim |xs | khs k

lim |xs | k fs k = lim |xs | khs k

s→∞

s→∞

= 0. Therefore if we put Ax =

∑ xs fs , then, A is a well-defined operator satisfying Ahs = fs .

s∈N

The second condition of the theorem implies that if we put B = C − A, then kBk = kC − Ak = kA −Ck < kCk , from which we deduce that kAk = kCk . Now

−1

BC ≤ kBk C−1 < kCk C−1 = 1, by assumption. Therefore, the operator AC −1 is such that

1 − AC −1 = BC −1 < 1. We can apply Theorem 2 to AC −1 and find that it is invertible. Since C is also invertible, it follows that A is invertible. Moreover, it is not difficult to show that A is isometric, and hence, ( fs )s∈N is an orthogonal base for Eω . The next corollary is a generalization of Corollary 2. Corollary 3. Let (hs )s∈N ⊂ Eω be an orthogonal basis, ( fs )s∈N ⊂ Eω a sequence of vectors and ζ a non-zero element of K satisfying: k fs − ζhs k < |ζ| , khs k s∈N

sup

then, ( fs )s∈N is an orthogonal base for Eω .

Non-Archimedean Bounded Linear Operators Proof. In Theorem 6, we take the matrix C to be the diagonal matrix C =

∑ cts

31 0 hs ⊗ ht

t,s∈N

with cts = 0 if t 6= s and ctt = ζ for all t ≥ 0. It is clear that for any s, Chs = ζhs , C is

invertible, C−1 hs = ζ−1 hs , kCk = |ζ| and C−1 = |ζ|−1 . Moreover k fs −Chs k k fs − ζhs k = sup khs k khs k s∈N s∈N < |ζ|

sup

= kCk . Corollary 4. Let (hs )s∈N be an orthogonal base for Eω . If (gs )s∈N is a sequence of vectors |hhk , gs i| kgs k = 0 for each k ∈ N, and that, of Eω satisfying: lim s→∞ |ωs | khs − Shs k < 1, sup |ωs |1/2 s∈N where Shk =

hhk , gs i gs for each k ∈ N, then (Shs )s∈N is an orthogonal base of Eω . s∈N ωs

∑

Remark 6. (1) Note that S defined above is a linear operator on Eω . |hhk , gs i| kgs k = 0 implies that (2) The assumption, lim s→∞ |ωs | Shk = is well-defined.

hhk , gs i gs , ∀k ∈ N s∈N ωs

∑

khs − Shs k (3) The operator S is isometric, by using sup |ωs |1/2 s∈N

< 1.

Proof. It suffices to put C = I, and fs = Shs in Theorem 6. 3.4.1 Example We illustrate Corollary 3 with the following example: Let p be a prime, K = Q, ωs = 1 ps , |ωs | = s . p As an orthogonal basis for Eω we use the canonical orthogonal basis (es )s∈N and recall 1 that kes k = |ωs |1/2 = s/2 . Let ς be such that |ς| = 1 and let p gs = (u − ς) es + p1+s

∑

et , ∀s ∈ N.

t∈N,t6=s

1 . To achieve this choice of u we do the p1+s following: ς is a p-adic unit which can be written in K as We choose u such that |u| = 1 and |u − ς| =

32

Toka Diagana ς = a0 + a1 p + a2 p2 + .... + as ps + as+1 ps+1 + ....,

where 1 ≤ a0 ≤ p − 1 and for k 6= 0, 0 ≤ ak ≤ p − 1. Setting u = a0 + a1 p + a2 p2 + .... + as ps it follows that |u| = 1 and u − ς = −(as+1 ps+1 + ....), and hence |u − ς| =

1 p1+s

.

Clearly, kgs k = max |u − ς||ωs |1/2 , 1

= max = max =

Thus

kgs k 1/2

|ωs |

=

1 p1+s/2

1 p1+s

p1+s+s/2 1 p1+s+s/2

, sup i6=s

,

1 p1+s

sup |ωi |1/2

!

i6=s

1

!

p1+s+i/2

1 p1+s

(even if s = 0).

, and

sup s

kgs k 1/2

|ωs |

= sup s

1 p1+s/2

=

1 < |ς| = 1. p

In summary, if we let fs = gs + ζes = ues + p1+s ∑ ei , then i6=s

k fs − ζes k kgs k < |ζ| . = sup 1/2 kes k s∈N s∈N |ωs |

sup

3.5 Hilbert-Schmidt Operators 3.5.1 Basic Definitions As in the classical setting, Hilbert-Schmidt operators in non-Archimedean Hilbert spaces are bounded linear operators satisfying a convergence property. Surprisingly, such convergence depends on the base. Definition 18. A bounded linear operator A : Eω 7→ Eω is a p-adic Hilbert-Schmidt operator if:

Non-Archimedean Bounded Linear Operators

Q (A) :=

"

∑

t∈N

kAet k ket k

2 #1/2

33

< ∞,

where (et )t∈N is the canonical orthogonal base for Eω . We denote by B2 (Eω ) the collection of all Hilbert-Schmidt operators on Eω . Remark 7. (i) If ket k = 1 for all t ∈ N, i.e., if (et )t∈N is an orthonormal base, we retrieve the definition of a Hilbert-Schmidt operator as in the classical context. Remark 8. In contrast with the classical setting, the definition of a Hilbert-Schmidt operator depends on the base. To find out, let p be an odd prime and set K = (Q p , | · |). Then | 12 | = |2| = 1. Let ωi = 1 for i = 0, 1, 2, ... and consider the corresponding p-adic Hilbert space Eω . Let (ei )i∈N be the canonical base for Eω and let fi = ei for i > 1 and f0 = e0 + e1 , and f1 = e0 − e1 . It can be shown that ( fi )i∈N is an orthogonal base for Eω . Let A : Eω 7→ Eω be the projection onto the one-dimensional space spanned by e0 : Ae0 = e0 , Aei = 0 for all i ≥ 1. It can be easily shown that A is a Hilbert-Schmidt operator and that 1=

kAes k2 kA fs k2 = 6 = 2. ∑ ∑ 2 s∈N |ωs | s∈N k fs k

Definition 19. Let A be a Hilbert-Schmidt operator on Eω . The real number

Q (A) :=

"

∑

s∈N

kAes k kes k

2 #1/2

is then well-defined and is called the Hilbert-Schmidt norm of A. Theorem 7. Let A ∈ B2 (Eω ), then the following statements hold true: (1) A has an adjoint denoted A∗ ; (2) Q (A) = Q (A∗ ) and A∗ ∈ B2 (Eω ); (3) kAk ≤ (Q (A)). Remark 9. In view of the above, B2 (Eω ) ⊂ B0 (Eω ). Proof. Let A = ∑ats e0s ⊗ et be a Hilbert-Schmidt operator. We first show that the boundt,s

edness is a direct consequence of the convergence property. The convergence of the real series yields: kAes k2 = 0, lim s→∞ |ωs |

34

Toka Diagana kAes k2 and therefore, the sequence is bounded. But since |ωs | s∈N kAk = sup s

kAes k kAes k = sup , 1/2 kes k s |ωs |

hence kAk < ∞. |ats | = 0. Again, because of the convergence of s→∞ |ωs |1/2

(1) We need to prove that for all t, lim the real series, lim ( s→∞

Since

kAes k2 ) = 0. |ωs | kAes k = k ∑ ats et k = sup |ats ||ωt |1/2 , t

t

it follows that

|ats | = 0. s→∞ |ωs |1/2

∀t,

lim

∗ ∗ = ω−1 ω a , then: (2) If A∗ = ∑ ats e0s ⊗ et with ats s st t t,s

kA∗ (es )k2 Q (A ) = ∑ |ωs | s 2

∗

=∑ s

supt |ωt |−2 |ωs |2 |ast |2 |ωt | |ωs |

= ∑ sup s

t

|ast |2 |ωs | . |ωt |

|ats |2 |ωt | but if we consider the real matrix C = |ωs | s t |ats |2 |ωt | |ast |2 |ωs | t then its transpose is C = (cst ) with cst = . Moreover, (cts ) where cts = |ωs | |ωt | t all the entries in column s of C are the same as all the entries in row s of C , therefore On the other hand, Q 2 (A) =

sup t

∑ sup

|ats |2 |ωt | |ast |2 |ωs | = sup . |ωs | |ωt | t

Thus, Q 2 (A∗ ) = Q 2 (A) < ∞. Consequently, A∗ ∈ B2 (Eω ). (3) We have

Q 2 (A) = ∑ sup s

t

|ats |2 |ωt | |ωs |

|ats |2 |ωt | ≥ sup sup |ωs | s t = kAk2 . This completes the proof.

!

Non-Archimedean Bounded Linear Operators

35

Theorem 8. The following statements hold true: (1) let A, B ∈ B0 (Eω ) and suppose that A ∈ B2 (Eω ), then both AB and BA are in B2 (Eω ). (2) let A, B ∈ B2 (Eω ), then AB ∈ B2 (Eω ) and

Q (AB) ≤ Q (A) . Q (B). Remark 10. (a) From (1) above it follows that B2 (Eω ) is a two-sided ideal in B0 (Eω ); (b) From (2) above, we have that (B2 (Eω ), Q (·)) is a normed algebra. Proof. (1) Observe that kBA (es )k ≤ kBk kA (es )k , hence kA(es )k2 kBA (es )k2 ≤ kBk2 . . |ωs | |ωs | Since B ∈ B(Eω ) and A ∈ B2 (Eω ) it follows that BA ∈ B2 (Eω ). Now by Theorem 7, A∗ exists and is in B2 (Eω ), hence, by the first part, B∗ A∗ ∈ B2 (Eω ). Again by Theorem 7, AB = (B∗ A∗ )∗ ∈ B2 (Eω ). (2) This is straightforward, and hence left to the reader. The following is now clear. Theorem 9. Let A ∈ B0 (Eω ), then the following are equivalent (1) A ∈ B2 (Eω ); (2) for all U,V ∈ B0 (Eω ),UAV ∈ B2 (Eω ). Proof. To see that (2) ⇒ (1) choose U = V = I. In the space B2 (Eω ) we introduce a form, namely: For all A, B ∈ B2 (Eω ), hA, Bi := ∑ s

hAes , Bes i . ωs

(3.5.1)

This defines a symmetric, bilinear and non-degenerate form on B2 (Eω ). The relationship between the bilinear form defined above and the Hilbert-Schmidt norm is given by the Cauchy-Schwarz inequality (see Theorem 10 below). Theorem 10. If A, B ∈ B2 (Eω ), then |hA, Bi| ≤ Q (A) . Q (B). Proof. We first give a proof along a classical line: hAes , Bes i |< A, B >| = ∑ ω s s ≤∑

|hAes , Bes i| |ωs |

≤∑

kAes k kBes k , |ωs |

s

s

36

Toka Diagana

by the Cauchy-Schwarz inequality for the bilinear form on Eω . kAes k and v = (vs )s with vs = Now we consider the sequences u = (us )s with us = |ωs |1/2 kBes k . Since both A and B are in B2 (Eω ), then both u and v are in l 2 (N). By Holder’s |ωs |1/2 inequality uv = (us vs )s ∈ l 1 (N) and kAes k . kBes k |ωs | s 1/2 1/2 kAes k2 kBes k2 ≤ ∑ ∑ |ωs | |ωs | s s

|uv|1 = ∑

= Q (A) . Q (B). Combining, one obtains |hA, Bi| ≤ Q (A) . Q (B). Next one gives another proof along an ultrametric line: first, write A = ∑ats e0s ⊗ et t,s and B = ∑bts e0s ⊗ et : t,s

hAes , Bes i |hA, Bi| = ∑ ω s

s

kAes k kBes k ≤ sup |ωs | s ! ! kAes k kBes k . sup ≤ sup 1/2 1/2 s |ωs | s |ωs | kAes k kBes k = sup . sup s kes k s kes k = kAk kBk ≤ Q (A) . Q (B), by [Theorem 7, (3)]. Here are other properties that are reminiscent of the classical case. Proposition 19. Let A = ∑ats e0s ⊗ et and suppose that the expression ts

|ats |2 |ωt | ∑ |ωs | = C2 < ∞. t,s∈N Then A ∈ B2 (Eω ) and Q (A) ≤ C. Proof. We observe that ∀s, sup |ats |2 |ωt | ≤ ∑ |ats |2 |ωt | . Now t

t

Non-Archimedean Bounded Linear Operators

∑ s

37

kAes k2 1 = ∑ sup |ats |2 |ωt | |ωs | s∈N |ωs | t 1 2 ≤ ∑ ∑ |ats | |ωt | t s∈N |ωs | =

|ats |2 |ωt | ∑ |ωs | t,s∈N

= C2 . Therefore, Q 2 (A) ≤ C2 . Remark 11. In contrast with the classical case, strict inequality may occur in Proposition 19. Indeed, let K = (Q p , | · |), ωt = p−t , ats = pt , hence |ωt | = pt , |ats | = p−t . (1) ∀s, lim |ats | |ωt |1/2 = lim t

t

1 = 0. pt/2

(2)

∑ t,s

|ats |2 |ωt | pt = ∑ 2t+s |ωs | t,s p =∑ t,s 2

1 pt+s

= C > 1. On the other hand, kAes k2 1 |ats |2 |ωt | ∑ |ωs | = ∑ |ωs | sup t s p s 1 1 =∑ sup t s |ωs | t p 1 =∑ s |ωs | 1 = 1 1− p p . = p−1 p p2 p > 1 it follows that < . p−1 p − 1 (p − 1)2 Proposition 20. Let A = ∑ats e0s ⊗ et ∈ B(Eω ), furthermore, suppose But since

t,s

(1) |ωs | > 1, for all s ∈ N; (2) sup |ωs | = M < ∞; s

38

Toka Diagana

(3) ∑(sup |ats |)2 < ∞. s

t

Then A ∈ B2 (Eω ). Proof. We simply observe that kAes k2 = |ωs |

sup |ats | |ωt |

1/2

2

t

|ωs | 2 ≤ M sup |ats | . t

Remark 12. Note that Proposition 20 above is reminiscent of the properties of operators, which are of the ”Kernel” type. 3.5.2 Further Properties of Hilbert-Schmidt Operators We now explore properties of Non-Archimedean Hilbert-Schmidt operators which are suggested by their classical counterparts. Set c0 (E∗ω ) := {(At )t∈N ∈ (E∗ω )N : lim kAt k|ωt |1/2 = 0}. t→∞

Clearly c0 (E∗ω ) is a non-Archimedean Banach space over K when endowed the norm k(At )t∈N k = sup kAt k|ωt |1/2 . t∈N

Also, let |xt | < ∞}. 1/2 t∈N |ωt |

Fω := {(xt )t∈N ∈ KN : sup

In the same way Fω is a non-Archimedean Banach space over K with the norm k(xt )t k = sup t∈N

|xt | |ωt |1/2

.

In the context of the present situation we refer to c0 (E∗ω ) as the set of all sequences of elements in E∗ω which converge to 0, and to Fω as the set of bounded sequences of elements of K. Proposition 21. As Banach spaces over K, E∗ω is isomorphic to Fω . Proof. As in [65], the mapping A → (Aes )s gives the desired isomorphism.

Non-Archimedean Bounded Linear Operators

39

3.5.3 Completely Continuous Operators Definition 20. An operator A ∈ B(Eω ) is completely continuous if it is the (uniform) limit in B(Eω ) of a sequence of operators of finite rank. We denote by C(Eω ) the subspace of all completely continuous operators on Eω . Definition 21. For any operator A ∈ B(Eω ), we define As , the s − component of A by the formula: x ∈ Eω , Ax = (As x)s∈N . It is clear that As ∈ E∗ω . Proposition 22. The mapping A → (As )s is an isomorphism of the Banach space C(Eω ) onto the Banach space c0 (E∗ω ) . Proof. The proof is similar to that of [65, Proposition 4]. Theorem 11. Every Hilbert-Schmidt operator is completely continuous, i.e., B2 (Eω ) ⊂ C(Eω ). Proof. Let A be a Hilbert-Schmidt operator, then, by Theorem 7, the adjoint A∗ does exist and it is also a Hilbert-Schmidt operator, hence

∑ s

kA∗ es k2 < ∞. |ωs |

kA∗ es k = 0. s |ωs |1/2 ai j (e0j ⊗ ei ) be the standard representation of A and As the s-component

It follows that lim Let A =

∑

(i, j)∈N×N

of A. It is enough to show that lim kAs k|ωs |1/2 = 0. s

Observe |As x| x6=0 kxk

kAsk = sup = sup i

= sup i

Hence kAs k |ωs |1/2 = sup i

Therefore lims kAs k |ωs |1/2 = 0.

|As ei | kei k |asi | |ωi |1/2

|asi | |ωs |1/2 |ωi |1/2

.

=

kA∗ es k |ωs |1/2

.

40

Toka Diagana

3.5.4 Trace One important notion in the classical theory is that of trace. We now define the trace of an operator in the non-Archimedean context. Definition 22. [12] For A ∈ B0 (Eω ), we define the trace of A to be tr(A) :=

hAes , es i s∈N hes , es i

∑

(3.5.2)

if the sum converges in K and where (es )s∈N is the canonical orthogonal base for Eω . We denote by B1 (Eω ) the collection of operators which have traces1 . Remark 13. (1) The convergence in K of the series in (3.5.2) is equivalent to hAes , es i = 0. lim s→∞ hes , es i (2) Since A ∈ B0 (Eω ), its adjoint A∗ exists and tr(A) = tr(A∗ ). Remark 14. Let A be a Hilbert-Schmidt operator. Does the definition of tr(A) dependent of the canonical orthogonal base (es )s∈N ? 3.5.5 Examples Throughout this subsection, we suppose that K a complete non-Archimedean field, ω = (ωi )i∈N is a sequence of nonzero elements in K, and Eω is the corresponding nonArchimedean Hilbert space to ω = (ωt )t∈N . Example 19. Suppose that the ground field K = (Q p , | · |) and consider the linear operator A on Eω defined by ∞

Aes =

∑ ak,s ek ,

k=0

where

aks =

      

ps+k if k ≤ s 1 + p + p2 + .... + ps 0

We also require that: (1) |ωt | ≥ 1 for each t ∈ N, (2) sup|ωt | ≤ M for some M > 0. t∈N 1

An operator A whose trace does exists is also called nuclear.

if k > s.

Non-Archimedean Bounded Linear Operators

41

Proposition 23. Under assumptions (1)-(2) above, the operator A defined above is HilbertSchmidt on Eω . 1/2 p−(s+k) , hence Proof. For each s, |aks | |ωk |1/2 = p−(s+k) |ωk |1/2 p ≤M

∀s, lim |aks | |ωk |1/2 = 0. k

Therefore, A is well-defined. Now

1 ps+k | . |ωk | 2 kAes k = max | 2 s 0≤k≤s 1 + p + p + .... + p

2

and hence

2

ps+k kAes k ≤ M max | | 0≤k≤s 1 + p + p2 + .... + ps 2

Since |

2

,

.

ps+k | p = p−(k+s) it follows that 1 + p + p2 + .... + ps 2 ps+k 2 | p = M p−2s , kAes k ≤ M max | 0≤k≤s 1 + p + p2 + .... + ps

that is, kAes k2 ≤ M p−2s , |ωs | by (1). +∞

Since the series

∑ M p−2s converges, hence A ∈ B2 (Eω) .

s=0

Example 20. Suppose that the ground field K = Q p and that ωs = p−s for all s ∈ N. For integers m ≥ 1 and n ≥ 0 let A(m,n) =

∑

1

m n t,s∈N ωt ωs

e0s ⊗ et .

Proposition 24. Under previous assumptions, A(m,n) is a Hilbert-Schmidt on Eω .

2 1 Proof. For all s, A(m,n) es = sup . Now since |ωt | = pt → ∞ as t → ∞, 2m−1 2n |ω | |ω | t∈N t s 1 ≤ M and it follows that there exists a positive M such that sup 2m−1 i |ωi |

(m,n) 2

A es M ≤ . |ωs | |ωs |2n+1 Again, since |ωs | = ps → ∞ as s → ∞ it follows that the series M

M

∑ |ω |2n+1 = ∑ ps(2n+1) s

converges, hence A(m,n) ∈ B2 (Eω ).

s

s

42

Toka Diagana

Example 21. This example is a variation of the one produced in [24]. Let A be the operator on Eω defined by et Aes = λs es + ∑ . ω t6=s t Proposition 25. Suppose that |ωs | ≥ 1 for all s, and consider a sequence (λs )s∈N in (K, |.|) satisfying: |λs | ≥ |ωs |−1/2 for all s, and ∑ |λs |2 converges. Then A ∈ B2 (Eω ). s

Proof. Clearly, A ∈ B (Eω ). Moreover, for s 1 kAes k = max |λs | |ωs | , sup j6=s ω j 2

2

!

= |λs |2 |ωs | .

The result now follows from the fact that ∑ |λs|2 converges. s

Remark 15. In the case of K = (Q p , | · |), one can take ωs = p−3s and λs = ps .

3.6 Open Problems Problem 1. Interesting questions related to this chapter consists of developing a comprehensive spectral theory for self-adjoint bounded operators, normal operators, completely continuous operators, and Hilbert-Schmidt operators, as it had been done in the classical setting. The following questions, raised in Diagana[20], still remain: Problem 2. Let T : Eω 7→ Eω be a bounded linear operator. Suppose that T does not have an adjoint. Can we perturb T so that the resulting operator has an adjoint? Problem 3. Let T be a bounded linear operator on Eω . Is it possible to decompose T as T = A + B, where A ∈ B0 (Eω ) and B does not have an adjoint? Problem 4. Characterize the collection of all bounded linear operators on Eω , which do not have adjoint. Problem 5. Let T be a Hilbert-Schmidt operator on Eω . Define the square root T 1/2 of T ? Does it exist an orthogonal basis (φt )t for Eω such that Tx =

∑

t∈N

µt

hx, φt i . φt , ωt

x ∈ Eω ,

where µt ∈ K, ∀t ∈ N?

3.7 Bibliographical Notes This chapter is entirely devoted to non-Archimedean bounded linear operators. Most of the results of this chapter can be found in Diagana et al. [4, 12, 17], and Diarra [24, 25].

Chapter 4

Non-Archimedean Unbounded Linear Operators 4.1 Introduction Let ω = (ωi )i∈N be a sequence of nonzero elements in a (complete) non-Archimedean field (K, | · |) and let (Eω , k · k, h·, ·i) be the corresponding non-Archimedean Hilbert space (Section 3.2). This chapter is devoted to unbounded linear operators on free Banach spaces (respectively, on non-Archimedean Hilbert spaces Eω ). As for non-Archimedean bounded linear operators, some of the results go along with the classical line and others deviate from it. For the most part, the statements of the results are inspired by their classical counterparts. However their proofs may depend heavily on the non-Archimedean nature of both Eω and the ground field K. As for non-Archimedean bounded operators on Eω , we shall see that there exist unbounded linear operators, which do not have adjoint. Both the closedness and the self-adjointness of those unbounded linear operators will be screened. To deal with both the closedness and the self-adjoiness of those linear operators, we will equip the direct sum Eω × Eω with both a complete non-Archimedean norm and a (non-Archimedean) hilbertian structure. As in the classical context, we consider an unitary operator Γ on Eω × Eω , which yields a remarkable description the adjoint A∗ of A in terms of A. As illustrations, several examples will be discussed. In particular, special attention is paid to the so-called (unbounded) diagonal operator, which will play a crucial role to diagonalizing self-adjoint linear operators (see Chapter 6). Throughout the rest of this chapter, (K, |·|) denotes a complete non-Archimedean valued field.

4.2 Basic Definitions Let (E, k · k) and (F, ||| · |||) be free Banach spaces over K and let (et )t∈N , (ht )t∈N denote the canonical orthogonal bases associated with E and F, respectively. The next definition is crucial throughout the rest of the book and is due to Diagana[17, 18].

44

Toka Diagana

Definition 23. An unbounded linear operator A from E into F is a pair (D(A), A) consisting of a subspace D(A) ⊂ E (called the domain of A) and a (possibly not continuous) linear transformation A : D(A) 7→ F. The domain D(A) contains the base (ei )i∈N and consists of all u = (ui )i∈N ∈ E such that Au = ∑ ui Aei converges in F, that is i∈N

  D(A) := {u = (ut )t∈N ∈ E : lim |ut | . |||Aet ||| = 0},   t→∞     A =

∑

ats e0s ⊗ ht , ∀s ∈ N, lim |at,s | . |||ht ||| = 0. t→∞

t,s∈N

The collection of those unbounded linear operators is denoted by U (E, F) and U (E) in the case when E = F. 4.2.1 Example Example 22. Let K = (Q p , | · |) and suppose that E = F = C(Z p , Q p ), is the nonArchimedean Banach space appearing in Examples 9 and 13 equipped with its corresponding sup norm. Let γ = (γt )t∈N : Z p 7→ Q p be an arbitrary sequence of functions, not necessarily continuous. Define  ∞   ) := {u(x) = ut ft (x) ∈ E : lim |ut | . kMγ ft k∞ = 0}, D(M  γ ∑  t→∞   t=0  ∞ ∞      Mγ u(x) := ∑ ut γt (x) ft (x), ∀ u(x) = ∑ ut ft (x) ∈ D(Mγ ). t=0

t=0

It can be easily checked that the linear operator Mγ is well-defined. Moreover, depending on the boundedness or not of the expression supkγt k∞ , the operator Mγ may or may not be t∈N

bounded1 . Throughout the rest of this chapter we take E = F = Eω with ω = (ωt )t∈N ⊂ K being a sequence of nonzero. We have previously seen that if A ∈ B(Eω ) then D(A) = Eω . Here, we will see that the domains of most of operators that we consider differ from Eω . As for bounded operators, there are elements of U (Eω ), which do not have adjoint. In the next definition, we state some necessary and sufficient conditions, which do guarantee the existence of the adjoint. 1

If sup kγt k∞ = ∞, then Mγ is an unbounded linear operator on C(Z p , Q p ). Otherwise, it is a bounded linear t∈N

operator.

Non-Archimedean Unbounded Linear Operators

45

4.2.2 Existence of the Adjoint Definition 24. A linear operator A = ∑ ats e0s ⊗ et in U (Eω ) is said to have an adjoint t,s

A∗ ∈ U (Eω ) if and only if: lim |ωs |−1/2 . |ats | = 0, ∀t ∈ N.

(4.2.1)

s→∞

Furthermore, under assumption (4.2.1), the adjoint A∗ of A is uniquely expressed by   D(A∗ ) := {v = (vt )t∈N ∈ Eω : lim |vt | kA∗ et k = 0},   t→∞   ∗   A =

∑

1

∗ 0 ats es ⊗ et ,

∗ ∀s ∈ N, lim |ats | |ωt | 2 = 0, t→∞

t,s∈N

∗ where ats = ωt−1 ωs ast .

Set U0 (Eω ) =

(

) |a | ts A = ∑ats e0s ⊗ et ∈ U (Eω ) : lim = 0, ∀t ∈ N . s→∞ |ωs |1/2 t,s

Clearly, B0 (Eω ) ⊂ U0 (Eω ). Remark 16. In the classical context, if B is an unbounded linear operator on a Hilbert space H, then (B∗ )∗ = B (B being the closure of B). In the non-Archimedean setting, it is not ∗ ∗ ) = difficult to check that if A = ∑ ats e0s ⊗ et ∈ U0 (Eω ), then A∗ ∈ U0 (Eω ). Moreover, (ats t,s∈N

ats , ∀t, s ∈ N, and hence (A∗ )∗ = A. 4.2.3 Examples of Unbounded Operators With no Adjoint Example 23. Set K = Q p , the field of p-adic numbers endowed with the p-adic absolute 1 3 value |·| and let ωt = p3t so that |ωt | 2 = p− 2 t . Define A = ∑ ats . (e0s ⊗et ) by its coefficients: t,s

 −s   p ats = 1   p−t

if t < s if t = s if t > s.

Proposition 26. The linear operator A defined above is in U(Eω ) and does not have an adjoint. Proof. Since ∀s, lim|ats ||ωt | 2 = lim pt p− 2 t = 0 it is then clear that A = ∑ ats . (e0s ⊗ et ) is 1

t

3

t>s

well-defined. Furthermore, ∀t, s ∈ N,

ts

46

Toka Diagana  3  p 2 (s−t)+s 1 |ats ||ωt | 2  λts := = 1 1  |ωs | 2  pt+ 32 s− 23 t

if t < s if t = s if t > s.

Consequently, ∀t, s, λt,s :=

|ats ||ωt |1/2 |ωs |1/2

 s  if t < s  p 1 if t=s ≥   p− 32 t if t > s.

Clearly, kAk := sup λt,s = ∞, hence A ∈ U (Eω ). t,s

To complete the proof we have to show that ∀t, lim s

|ats |

∀t, lim s

|ωs |

1 2

|ats | 1

6= 0. Indeed,

|ωs | 2

3

= lim ps p 2 s = ∞, s>t

hence the adjoint of A does not exist. Similarly, taking bts = p−t−s , for all t, s ∈ N, and ωt = p3t for each t ∈ N, one can easily check that: B = ∑ p−t−s e0s ⊗ et 6∈ U0 (Eω ). t,s∈N

4.3 Closed Linear Operators on Eω Let A ∈ U (Eω ). As in the classical setting, we define the graph of the linear operator A by

G (A) := {(x, Ax) ∈ Eω × Eω : x ∈ D(A)}. Definition 25. An operator A ∈ U (Eω ) is said to be closed if its graph is a closed subspace in Eω × Eω . The operator A is said to be closable if it has a closed extension. As in the classical theory of unbounded linear operators we characterize the closedness of an operator A ∈ U (Eω ) as follows: ∀ut ∈ D(A) such that ku − ut k 7→ 0 and kAut − vk 7→ 0 (v ∈ Eω ) as t 7→ ∞, then u ∈ D(A) and Au = v. Remark 17. Note that if A ∈ B(Eω ), then it is closed. Indeed since A is bounded, D(A) = Eω . Moreover if ut ∈ Eω such that ut 7→ u on Eω as t 7→ ∞, then by the boundedness of A it follows that Aut 7→ Au as t 7→ ∞, that is, (ut , Aut ) 7→ (u, Au) as t 7→ ∞ on Eω × Eω , hence G (A) is closed. Let C(Eω ) denote the collection of closed linear operators A ∈ U (Eω ) . Obviously, B(Eω ) ⊂ C(Eω ).

Non-Archimedean Unbounded Linear Operators

47

Remark 18. As in the classical setting, it is not difficult to see that if A ∈ C(Eω ) and if B ∈ B(Eω ), then A + B ∈ C(Eω ). Definition 26. If A, B are (possibly unbounded) linear operators on Eω such that D(A) ⊂ D(B) and Ax = Bx for each x ∈ D(A), then B is said to be an extension of A and write A ⊂ B. Let A and B be (possibly unbounded) linear operators on Eω ; their algebraic sum and product are respectively defined by   D(A + B) = D(A) ∩ D(B)    (A + B)u = Au + Bu, ∀u ∈ D(A) ∩ D(B),

and    D(AB) = {u ∈ D(B) : Bu ∈ D(A)}   (AB)u = A(Bu), ∀u ∈ D(AB).

The domain D(A) ∩ D(B) of the algebraic sum A + B has to be watched with care. Indeed, it may happen D(A) ∩ D(B) = {0}. In this event, A + B is trivial. Let A : D(A) 7→ Eω be an unbounded linear operator on Eω . Note that in addition to the topology of Eω , D(A) can be provided with another norm that we will call non-Archimedean graph norm defined by: kukD(A) = max(kuk, kAuk), ∀u ∈ D(A). It is clear that the non-Archimedean graph norm is stronger than the norm of Eω . Moreover, those two norms are equivalent if and only if A ∈ B(Eω ). Proposition 27. Let A : D(A) 7→ Eω be an unbounded linear operator on Eω . (D(A), k · kD(A) ) is a non-Archimedean Banach space if and only if A is closed.

Then

Proof. It is easy to see that (D(A), k · kD(A) ) is a non-Archimedean normed vector space. Suppose that A is closed. Let (ut )t∈N ⊂ D(A) be a Cauchy sequence. Therefore, for each ε > 0 there exists T0 such that: kut − us kD(A) = max(kut − us k, kAut − Aus k) ≤ ε whenever t ≥ s ≥ T0 . Consequently, both (ut )t∈N (Aut )t∈N are Cauchy sequences in Eω . Since Eω is complete, ut → u, Aut → ξ as t → ∞. Clearly, u ∈ D(A) and Au = ξ because of the closedness of A, and hence kut − ukD(A) 7→ 0 as t → ∞, that is, (D(A), k · kD(A) ) is complete. Now suppose that (D(A), k · kD(A) ) is complete and let (ut )t∈N ⊂ D(A) such that kut − uk and kAut − ξk → 0 as t → ∞. We want to show that u ∈ D(A) and Au = ξ. Clearly,

48

Toka Diagana kut − us kD(A) = max(kut − us k, kAut − Aus k) → 0 as t, s → ∞,

and hence (ut )t∈N is a Cauchy sequence in (D(A), k · kD(A) ). In view of the above, there exists v ∈ D(A) such that kut − vkD(A) = max(kut − vk, kAut − Avk) → 0 as t → ∞. In particular, kut − vk, kAut − Avk → 0 as t → ∞, and therefore u = v ∈ D(A) and Av = Au = ξ, because of the uniqueness of the limit. Definition 27. An operator A ∈ U (Eω ) is said to be symmetric if hAu, vi = hu, Avi for all u, v ∈ D(A). Definition 28. An operator A ∈ U0 (Eω ) is said to be self-adjoint if D(A) = D(A∗ ) and Au = A∗ u for each u ∈ D(A). Clearly, every self-adjoint operator is symmetric. We shall prove the following important theorem: Theorem 12. Let A ∈ U0 (Eω ). Then its adjoint A∗ is a closed linear operator. In particular, if A is self-adjoint, then it is closed. Let Γ be the bounded linear operator which goes from Eω × Eω into itself defined by: Γ(u, v) := (−v, u), ∀(u, v) ∈ Eω × Eω . Clearly, Γ satisfies: (i) Γ2 (u, v) = −(u, v), ∀(u, v) ∈ Eω × Eω ; (ii) Γ2 (M) = M for any subspace M of Eω × Eω . From (i) it follows that: Γ2 = −I (Γ is an unitary operator), where I is the identity operator of Eω × Eω . Theorem 13. If A ∈ U0 (Eω ), then G (A∗ ) = [ΓG (A)]⊥ , where [ΓG (A)]⊥ is the orthogonal complement of [ΓG (A)]. Proof. (Theorem 13). Since the adjoint A∗ of A does exist, one defines its graph by:

G (A∗ ) := {(x, A∗ x) ∈ Eω × Eω : x ∈ D(A∗)}. Thus for (x, y) ∈ D(A∗ ) × D(A) one has h(x, A∗ x), Γ(y, Ay)i2 = h(x, A∗ x), (−Ay, y)i2 = −hx, Ayi + hA∗ x, yi = 0,

Non-Archimedean Unbounded Linear Operators

49

hence G (A∗ ) ⊂ [ΓG (A)]⊥ . Conversely, if (x, y) ∈ [ΓG (A)]⊥ , ∀z ∈ D(A), then, 0 = h(x, y), (−Az, z)i2 = −hx, Azi + hy, zi. It follows that hAz, xi = hz, yi. Now by uniqueness of the adjoint, we obtain that: x ∈ D(A∗ ) and A∗ x = y, hence (x, y) ∈ G (A∗ ). Proof. (Theorem 12). Since the adjoint of A∗ of A does exist and that ΓG (A) is a subspace of Eω × Eω , then using Theorem 13 it follows that G (A∗ ) = [ΓG (A)]⊥ is closed, hence A∗ is closed.

4.4 Diagonal Operators on Eω Let ω = (ωt )t∈N ⊂ K be a sequence of nonzero elements and let Eω be its corresponding non-Archimedean Hilbert space. Let (λt )t∈N ⊂ K be a sequence of elements such that lim |λt | = ∞.

(4.4.1)

t→∞

Define the diagonal operator A ∈ U (Eω ) by D(A) = {x = (xt ) ⊂ K : lim |λt | . |xt | . ket k = 0}, t

and Ax =

∑ λt xt et ,

for each x =

t∈N

∑ xt et ∈ D(A).

t∈N

Proposition 28. Under assumption (4.4.1), the diagonal operator A defined above is a (unbounded) self-adjoint operator on Eω . Furthermore, ρ(A) = {λ ∈ K : λ 6= λt , ∀t ∈ N}, and 1 −1 k(A − λ) k = sup t∈N |λt − λ| for each λ ∈ ρ(A). Proof. First of all, let us make sure that the operator A is well-defined. For that, note that |att | = |λt | and that |ats | = 0 if t 6= s, and lim |ats | |ωt |1/2 = lim |ats | |ωt |1/2 = 0, t

t>s

and hence A is well-defined. |ats | |ωt |1/2 Now = |λt | if t = s and 0 if t 6= s. It follows that |ωs | kAk := sup t,s

|ats | |ωt |1/2 = sup |λt | = ∞, |ωs | t

50

Toka Diagana

and hence A ∈ U (Eω ). Let us show that the adjoint A∗ of A does exist. This is actually obvious since ∀t ∈ N, lim |ats | |ωs |−1/2 = lim |ats | |ωs |−1/2 = 0. s

s>t

Clearly, the adjoint A∗ is defined by A∗ = ∑ bts e0s ⊗ et , where bts = wt−1 ws ast = ats for ts

all t, s ∈ N, and hence A = A∗ . To complete the proof we have to compute elements of the resolvent ρ(A) of A. We want to find all λ ∈ K such that (A − λI)x = y,

(4.4.2)

where x = ∑xt et ∈ D(A) = D(A − λI) and y = ∑yt et ∈ Eω . t

t

Considering (4.4.2) on (et )t∈N and using the fact A is self-adjoint it follows that ∀t ∈ N, (λt − λ) . het , xi = het , yi. Equivalently, ∀t ∈ N, (λt − λ) . ωt xt = ωt yt . For all λt 6= λ, (4.4.2) has a unique solution x. Moreover, yt et . x = (A − λ)−1y = ∑ t∈N λt − λ

(4.4.3)

(4.4.4)

Let us show that x = (A − λ)−1 y given above is well-defined. For that it is sufficient to prove that |yt | ket k = 0. lim t→∞ |λt − λ| 1 is bounded, and hence Using (4.4.1) it easily follows that the sequence |λt − λ| t∈N |yt | ket k = 0. t→∞ |λt − λ| lim

It remains to find conditions on λ so that x defined above belongs to D(A). For that, it is sufficient to show that lim

t→∞

|yt | |λt | kAet k = lim |yt | ket k = 0. t→∞ |λt − λ| |λt − λ|

Indeed, since lim |yt | ket k = 0, t→∞

|yt | kAet k |λt − λ| |λt | . lim |yt | ket k ≤ lim t→∞ | |λt | − |λ| | t = 0.

0 ≤ lim

t→∞

Non-Archimedean Unbounded Linear Operators From (4.4.4) it follows that for each t ∈ N, k(A − λ)−1 et k =

51

kes k , in other words, |λt − λ|

1 , k(A − λ) k = sup t∈N |λt − λ| −1

and hence (A − λ)−1 ∈ B(Eω ). In summary, the resolvent ρ(A) of A given by: ρ(A) = {λ ∈ K : λ 6= λt , ∀t ∈ N}. Remark 19. Note that D(A), the domain of the diagonal operator A may or may not be equal the whole Eω . Indeed, take, for each t ∈ N, µt ∈ K − {0}, ωt = µt2 , and x˜ = (x˜t )t∈N where 1 x˜t = for all t ∈ N. Clearly, x˜ ∈ Eω since λt µt lim |x˜t |ket k = lim

t→∞

1

t→∞ |λt |

= 0,

by (4.4.1). Meanwhile, one can easily see that x˜ 6∈ D(A) since lim |x˜t | . |λt | . ket k = 1 6= 0.

t→∞

4.5 Open Problems Problem 1. Interesting questions related to this chapter consists of developing a comprehensive spectral theory for self-adjoint and normal (unbounded) linear operators. Problem 2. Let T : D(T ) ⊂ Eω 7→ Eω be a unbounded linear operator. Suppose that T does not have an adjoint. How can we perturb T so that the resulting operator has an adjoint? Problem 3. Let T be an unbounded linear operator on Eω . Is it possible to decompose T as: T = A + B, where A ∈ U0 (Eω ) and B does not have an adjoint? Problem 4. Characterize (spectrally) the collection of all unbounded linear operators on Eω , which do not have adjoint? Problem 5. Let A, B be unbounded linear operators on Eω . Find necessary and sufficient conditions so that A + B ∈ C(Eω ).

4.6 Bibliographical Notes This chapter is entirely devoted to non-Archimedean unbounded linear operator. Most of results (published or not) of this chapter are due to Diagana[17, 18].

Chapter 5

Non-Archimedean Bilinear Forms 5.1 Introduction In the classical setting, Bounded and unbounded sesquilinear forms on Hilbert spaces play an crucial role in several fields such as quantum mechanics, mathematical physics, variational analysis, symplectic geometry, see, e.g., de Bivar-Weinholtz and Lapidus [5], Diagana [19], Johnson and Lapidus [34], and Kato[35]. The present chapter examines a preliminary work of the author on non-Archimedean counterparts of the classical bilinear forms on Hilbert spaces. To do so, we first introduce and study bounded and unbounded (symmetric) bilinear forms on Eω × Eω ; we then call those forms non-Archimedean bilinear forms. Secondly, we emphasis on the closedness of these bilinear forms in connection with the associated non-Archimedean quadratic forms as it had been done in the classical setting in Kato [35, Chapter VI, p. 313]. In particular, some sufficient conditions for the closedness of the form sum of the so-called quadratically disjoint closed bilinear forms, are given. In addition to the above, we also discuss the construction of non-Archimedean Hilbert spaces through bilinear forms. At the end of this chapter we discuss on a new representation theorem, which basically says that under some suitable assumptions, each non-degenerate (unbounded) bilinear form on Eω × Eω is representable by a linear operator A under some. Furthermore, when such a bilinear form is symmetric, then the operator A is self-adjoint. If ω = (ωt )t∈N is a sequence of nonzero elements in K, then we let Eω denote its corresponding non-Archimedean Hilbert space and (et )t∈N denotes the canonical orthogonal base associated with Eω .

5.2 Basic Definitions 5.2.1 Continuous Linear Functionals on Eω Definition 29. A linear functional ϕ : Eω 7→ K is said to be continuous if there exists K ≥ 0 such that

54

Toka Diagana |ϕ(u)| ≤ K . kuk, u ∈ Eω .

(5.2.1)

The smallest K such that (5.2.1) holds is called the norm of the continuous linear functional ϕ and is defined by |ϕ(u)| |kϕk| = sup . kuk u6=0 The space of all continuous linear functionals on Eω is denoted E∗ω and called the (topological) dual of Eω . The space (E∗ω , |k · k|) is a Banach space over K. Proposition 29. Let ϕ : Eω 7→ K be a continuous linear functional. Then its norm |kϕk| can be explicitly expressed as |ϕ(ei )| . |kφk| = sup kei k i∈N |ϕ(ei )| , by the definition of the norm |kϕk|. Proof. Obviously, |kφk| ≥ sup kei k i∈N Now let u 6= 0, ∞ |ϕ(u)| = ∑ ϕ(ei ) ui i=0 ≤ sup (|ϕ(ei )| . |ui |) i∈N |ϕ(ei )|(|ui | . kei k) = sup kei k i∈N |ϕ(ei )| ≤ kuk . sup , kei k i∈N

and hence

|ϕ(ei )| . |kϕk| ≤ sup kei k i∈N

One completes the proof by combining the first and the latest inequalities. The next theorem is a non-archimedean version of the well-known Riesz representation theorem [35]. Theorem 14. Let ϕ : Eω 7→ K be a linear functional such that kϕ(ei )k = 0. i→∞ kei k lim

Then there exists a unique u0 ∈ Eω such that ϕ(u) = hu, u0 i, for all u ∈ Eω . Moreover, |kϕk| = ku0 k.

(5.2.2)

Non-Archimedean Bilinear Forms

55

kϕ(ei )k < ∞. i∈N kei k Let u = ∑ ui ei ∈ Eω . Now, φ(u) = ∑ ui ϕ(ei ) is well-defined. Indeed, since u ∈ Eω ,

Proof. Obviously, (5.2.2) yields ϕ is continuous as |kϕk| = sup i∈N

i∈N

lim |ui |kei k = 0, and hence

i→∞

kϕ(ei )k = 0, i→∞ kei k

lim |ui ϕ(ei )| ≤ kuk . lim

i→∞

by (5.2.2).

φ(ei ) ei . Using (5.2.2), one can easily see that u0 ∈ Eω . Moreover, ωi i∈N ϕ(u) = hu, u0 i for each u ∈ Eω . Suppose that there exists another v0 ∈ Eω such that ϕ(u) = hu, v0 i for each u ∈ Eω . Then, hu0 − v0 , ui = 0 for each u ∈ Eω , that is, u0 − v0 ⊥ Eω . In particular, hu0 − v0 , ei i = 0 for each i ∈ N, that is, all coordinates of u0 − v0 in the canonical base (ei )i∈N of Eω are zero, and hence u0 = v0 . Now

φ(ei ) kϕ(ei )k

ku0 k := sup

ωi ei = sup kei k = |kϕk|. i∈N i∈N Now, set u0 =

∑

Definition 30. A mapping φ : Eω × Eω 7→ K is called a non-Archimedean bilinear form if u 7→ φ(u, v) is linear for each v ∈ Eω and v 7→ φ(u, v) linear for each u ∈ Eω . One can easily see that if φ : Eω × Eω 7→ K is a non-Archimedean bilinear form over Eω × Eω , then, for all u = (ut )t∈N , v = (vt )t∈N ∈ Eω , φ(u, v) =

∞

∑ σts ut . vs ,

∀s ∈ N, lim |ut ||σts |1/2 = 0,

t,s=0

t→∞

(5.2.3)

where σts = φ(et , es ) for all t, s ∈ N. 5.2.2 Bounded Bilinear Forms on Eω × Eω Definition 31. A non-Archimedean bilinear form φ : Eω × Eω 7→ K is said to be bounded if there exists M ≥ 0 such that |φ(u, v)| ≤ M . kuk . kvk

(5.2.4)

for all u, v ∈ Eω . The smallest M such that (3.3.1) holds is called the norm of the bilinear form φ and is defined by |φ(u, v)| . kφk = sup u,v6=0 kuk . kvk

56

Toka Diagana

Example 24. Let A : Eω 7→ Eω be a bounded linear operator. Setting Φ(u, v) = hAu, vi for all u, v ∈ Eω , where h·, ·i is the inner product given in (2.3.2), it is then clear that Φ is a bounded nonArchimedean bilinear form over Eω × Eω . Proposition 30. Let φ : Eω × Eω 7→ K be a bounded bilinear form. Then its norm kφk can be explicitly expressed as |φ(ei , e j )| . kφk = sup i, j∈N kei k . ke j k |φ(ei , e j )| , is a straightforward consequence of the Proof. The inequality, kφk ≥ sup i, j∈N kei k . ke j k definition of the norm kφk of φ. Now suppose u, v 6= 0. In view of the above, one has ∞ |φ(u, v)| = ∑ φ(ei , e j ) ui v j i, j=0 ≤ sup (|φ(ei , e j )| . |ui | . |v j |) i, j∈N

|φ(ei , e j )|(|ui | . kei k) (|v j | . ke j k) kei k . ke j k i, j∈N |φ(ei , e j )| ≤ kuk . kvk . sup , i, j∈N kei k . ke j k = sup

and hence kφk ≤ sup i, j∈N

|φ(ei , e j )| . kei k . ke j k

One completes the proof by combining the first and the latest inequalities. 5.2.3 Unbounded Bilinear Forms on Eω × Eω We now introduce non-Archimedean analogues of the so-called unbounded bilinear forms. Definition 32. A mapping Ψ : D(Ψ) × D(Ψ) ⊂ Eω × Eω 7→ K is called a non-Archimedean (unbounded) bilinear form if u 7→ Ψ(u, v) is linear for each v ∈ D(Ψ) and v 7→ Ψ(u, v) linear for each u ∈ D(Ψ), where  D(Ψ) := {u = (ut )t∈N ∈ Eω : lim |ut | . |Ψ(et , et )|1/2 = 0},   t→∞       Ψ(u, v) =

∞

∑ σts ut vs ,

t,s=0

∀s ∈ N, lim |ut | . |σts |1/2 = 0 t→∞

for all u, v ∈ D(Ψ), where σts = Ψ(et , es ) for all t, s ∈ N.

Non-Archimedean Bilinear Forms

57

The subspace D(Ψ) will be called the domain of the non-Archimedean bilinear form ψ. One says that Ψ is densely defined if D(Ψ) is dense in Eω . Example 25. Let A : D(A) ⊂ Eω 7→ Eω be an unbounded linear operator. One can easily check that, ψ(u, v) = hAu, Avi, for u = (ut )t∈N , v = (vt )t∈N ∈ D(A) is a non-Archimedean bilinear form, which can be explicitly expressed by ψ(u, v) =

∞

∑ σts ut . vs

t,s=0

where σts = hAet , Aes i, t, s = 0, 1, 2, ... Clearly, ψ given above is well-defined. Indeed, for each u = (ut )t∈N ∈ D(A), |ut | . |hAet , Aet i|1/2 ≤ |ut | kAet k 7→ 0 as t 7→ ∞, by u = (ut )t∈N ∈ D(A). Moreover, D(A) ⊂ D(ψ). We now consider an example of bilinear forms corresponding to diagonal operators. Let (λt )t∈N ⊂ K be a sequence of terms such that lim |λt | = ∞. Define the operator A by t→∞

 D(A) = {u = (ut )t∈N : lim |λt ||ut |ket k = 0},   t→∞  

∞ ∞     Au = ∑ λt ut et for each u = ∑ ut et ∈ D(A). t=0

t=0

One can easily check that A is well-defined and is an unbounded linear operator on Eω . Define ψ(u, v) = hAu, vi for each u, v ∈ D(A) by ∞

ψ(u, v) = ∑ ωt λt ut vt

(5.2.5)

t=0

∞

for all u = ∑ ut et , v = t=0

∞

∑ vt et ∈ D(A).

t=0

Clearly, ψ is well-defined. Indeed, |ωt ||λt ||ut ||vt | = (|λt ||ut |ket k)(|vt |ket k) for each t ∈ N, and hence lim |ωt ||λt ||ut ||vt | = 0, by u ∈ D(A) and v ∈ D(A) ⊂ Eω . Next, it t→∞ is straightforward to check that ψ is a non-Archimedean bilinear form. If ψ : D(ψ) × D(ψ) ⊂ Eω × Eω 7→ K is a non-Archimedean bilinear form, then q denotes the non-Archimedean quadratic form associated with ψ and is defined by q(u) = ψ(u, u) for each u ∈ D(ψ) = D(q). Thus one can easily check that the relationship between the quadratic form q and the non-Archimedean bilinear form ψ can be expressed as 4 . ψ(u, v) = q(u + v) − q(u − v), ∀u, v ∈ D(ψ).

(5.2.6)

58

Toka Diagana Furthermore, 2 . [q(u) + q(v)] = q(u + v) + q(u − v)

(5.2.7)

for all u, v ∈ D(ψ). Let φ, ψ be p-adic bilinear forms. One defines the p-adic form sum of these nonArchimedean bilinear forms by setting    D(φ + ψ) = D(φ) ∩ D(ψ),   (φ + ψ)(u, v) = φ(u, v) + ψ(u, v)

for all u, v ∈ D(φ) ∩ D(ψ). Similarly, D(λ . φ) = D(φ) and (λ . φ)(u, v) = λ . φ(u, v) for all λ ∈ K − {0} and u, v ∈ D(φ).

5.3 Closed and Closable non-Archimedean Bilinear Forms Let a be a non-Archimedean (unbounded) bilinear form. As in the classical context one defines the so-called φ-convergence as follows. Definition 33. A sequence (xt )t∈N ∈ Eω is said to be φ-convergent to x ∈ Eω if (xt )t∈N ∈ D(φ), xt 7→ x in Eω and q(xt − xs ) 7→ 0 in K as t, s 7→ ∞. The φ-convergence of a sequence (xt )t∈N ∈ Eω to x ∈ Eω will be denoted by x = φ − lim xt . As in the classical context, note that the φ-limit, x, appearing in Definit→∞

tion 33, may or may not belong to D(φ). Example 26. Consider the non-Archimedean bilinear form φ (respectively, its corresponding quadratic form q) given in Example 25 with K = Q p , the field of p-adic numbers equipped with the p-adic absolute value | · |. (p ≥ 2 being a prime.) Suppose ωt = pt for each t ∈ N and that λt = p−t for each t ∈ N. Clearly, lim |λt | = ∞, and t→∞

    

t

D(A) = {u = (ut )t∈N ∈ Eω : lim p 2 |ut | = 0}, t→∞

∞ ∞   t   Au = ∑ p ut et for each u = ∑ ut et ∈ D(A), and t=0

t=0

∞

∞

t=0

t=0

q(u) = ∑ ut2 for each u = ∑ ut et ∈ D(A). Here, D(φ) = D(q) = {u = (ut )t∈N ∈ Eω : lim |ut | = 0}. t→∞

Non-Archimedean Bilinear Forms

59

t

Clearly, D(A) ⊂ D(φ) since |ut | ≤ p 2 |ut | 7→ 0 as t 7→ ∞ whenever u = (ut )t∈N ∈ D(A). Now consider the sequence defined by yts = p3t+s for each t, s ∈ N. One can easily check that (ys )s∈N ∈ D(A), yts 7→ 0 as s 7→ ∞ for each t ∈ N, and q(ys − yr ) = (ps − pr )2 ∑ p6t 7→ 0 in Q p as s, r 7→ ∞ t∈N

and hence φ − lim ys = 0. s→∞

Definition 34. A non-Archimedean bilinear form φ is said to be closed if x = φ − lim xt t→∞

yields x ∈ D(φ) and that q(xs − x) 7→ 0 as s 7→ ∞.

Theorem 15. Let a : D(φ)×D(φ) 7→ K be a bounded non-Archimedean bilinear form. Then φ is closed if and only if D(φ) is a closed subspace of Eω . Proof. Suppose that φ is a closed bounded non-Archimedean bilinear form, i.e., φ is closed and there exists M ≥ 0 such that |φ(u, v)| ≤ M . kuk . kvk, ∀(u, v) ∈ D(φ) × D(φ). Let (us )s∈N ∈ D(φ) such that us 7→ u in Eω . So we have to show that u ∈ D(φ). Now from the boundedness of φ and the fact that (us )s∈N is a Cauchy sequence in Eω (us 7→ u in Eω ) it is clear that q(ut − us ) 7→ 0 as t, s 7→ ∞ where q is the quadratic form associated with φ, and hence u = φ − lim us . Using the closedness of φ it follows that u ∈ D(φ), and therefore s→∞

D(φ) is a closed subspace of Eω . Conversely, suppose that D(φ) is a closed subspace of Eω and let u = φ − lim us . Thus s→∞

(us )s∈N ∈ D(φ), us 7→ u in Eω as s 7→ ∞, and q(ut − us ) 7→ 0 as t, s 7→ ∞. Since D(φ) is closed, then u ∈ D(φ). Again, by the boundedness of φ and the fact us 7→ u in Eω as s 7→ ∞ it follows that q(ut − u) 7→ 0 as t 7→ ∞, and hence φ is closed. 5.3.1 Closedness of the Form Sum Definition 35. Let φt (t = 1, 2, ...n) be n non-Archimedean bilinear forms and let qt (t = 1, ...n) be the associated quadratic forms, respectively. One says that the sequence of nonArchimedean bilinear forms (φt )t=1,...n is quadratically disjoint if its satisfies 6 |q2 (u)| 6= ... 6= |qn (u)| |q1 (u)| =

(5.3.1)

for each 0 6= u ∈ D(φ1 ) ∩ D(φ2 ) ∩ ... ∩ D(φn). Theorem 16. Let (φt )t=1,...n be a (finite) sequence of bilinear forms such that each φt (t = 1, 2, ..., n) is closed. If (5.3.1) holds, then the form sum φ = φ1 + φ2 + .... + φn with domain D(φ) = D(φ1 ) ∩ D(φ2 ) ∩ ... ∩ D(φn) is closed.

60

Toka Diagana

Proof. Suppose φ − lim us = u for some sequence (us )s∈N . Clearly, if the sequence (us )s∈N s→∞

is constant, then there is nothing to prove. Now suppose that (us )s∈N is a nonconstant sequence. Consequently, q1 (ut − us ) + q2 (ut − us ) + ... + qn(ut − us) 7→ 0 as t, s 7→ ∞. Clearly, among the natural integers, 1, 2, 3, ...,n, there exists k0 such that |q1 (ut − us ) + q2 (ut − us ) + ... + qn (ut − us )| = |qk0 (ut − us )|. Namely, |qk0 (ut − us )| = max(|q1 (ut − us )|, |q2 (ut − us )|, ..., |qn (ut − us )|), by using (5.3.1) and the fact K is non-Archimedean. And hence |qk0 (ut − us )| 7→ 0 as t, s 7→ ∞. Consequently, ak0 − lim us = u, and therefore u ∈ D(φk0 ) and qk0 (us − u) 7→ 0 as s 7→ ∞. s→∞

Using the fact that |qr (ut − us )| < |qk0 (ut − us )| for each r = 1, 2, ...n with r 6= k0 it easily follows that u ∈ D(φr ) and qr (us − u) 7→ 0 for each k = 1, 2, ..., n with r 6= k0 as s 7→ ∞. In summary, u ∈ D(φ) and q(us − u) 7→ 0 as s 7→ ∞, where q = q1 + q2 + ... + qn, and therefore φ is closed. Remark 20. (1) Theorem 6 still holds when the quadratically disjoint assumption, Eq. (5.3.1), is replaced by the strictly increasing assumption given by |q1 (u)| < |q2 (u)| < ... < |qn (u)|. for each 0 6= u ∈ D(φ1 ) ∩ D(φ2) ∩ ... ∩ D(φn). (2) In the classical context one makes use of both the closedness and the positiveness of the real part of each form φt (t = 1, 2, ..., n) to deduce the closedness of the form sum φ. Here, the situation is much more tricky since there is no notion of positiveness in a general non-Archimedean field K. Corollary 5. Let φ : D(φ) × D(φ) ⊂ Eω × Eω 7→ K be a closed non-Archimedean bilinear form. Suppose that ∞ ∞ ∑ ωt ut2 6= |φ(u, u)| for all nonzero u = ∑ ut et ∈ D(φ). t=0 t=0 Then the form sum ψ defined by ψ(u, v) := hu, vi + φ(u, v) for all u, v ∈ D(φ) is closed.

Non-Archimedean Bilinear Forms 61 ∞ ∞ Proof. Suppose that ∑ ωt ut2 < |φ(u, u)| each u = ∑ ut et ∈ D(φ) with u 6= 0. (when t=0 t=0 ∞ ∑ ωt ut2 > |φ(u, u)|, one follows along the same lines as the case being treated.) Set t=0 ξ(u, v) = hu, vi for all u, v ∈ Eω where h·, ·i is the inner product of Eω defined in (2.3.2). Clearly ξ is bounded, by the Cauchy-Schwarz inequality. Now since D(ξ) = Eω , then ξ is closed, by Theorem 15. One completes the proof by setting ψ = φ + ξ and apply it to Theorem 6. 5.3.2 Construction of a non-Archimedean Hilbert Space Using Quadratic Forms Let φ : D(ζ) × D(ζ) ⊂ Eω × Eω 7→ K be a non-degenerate non-Archimedean bilinear form. Define hu, viζ := hu, vi + ζ(u, v)

(5.3.2)

kuk2q := max(|q(u)|, kuk2 )

(5.3.3)

for all u, v ∈ D(ζ), and

for each u ∈ D(ζ), where q is the quadratic form associated with the non-Archimedean bilinear form ζ. One requires the following assumption |q(u + v)| ≤ max(|q(u)|, |q(v)|)

(5.3.4)

for all u, v ∈ D(ζ). One can easily check that h·, ·iζ is symmetric, non-degenerate form on D(ζ) × D(ζ) with values in K. Under assumption (5.3.4) it is clear that k · kq is a non-Archimedean norm on D(ζ). Indeed, for all u, v ∈ D(ζ) ku + vk2q = max(|q(u + v)|, ku + vk2 ) ≤ max max(|q(u)|, |q(v)|), max(kuk2 , kvk2 ) = max |q(u)|, |q(v)|, kuk2 , kvk2 ≤ max(kuk2q , kvk2q , kuk2q , kvk2q ) = max(kuk2q , kvk2q ). In other words, ku + vkq ≤ max(kukq , kvkq ), and hence k.kq is a non-Archimedean norm over D(ζ). Let Eq denote the normed non-Archimedean space (D(ζ), k · kq ). Theorem 17. Let ζ be a non-degenerate non-Archimedean bilinear form over D(ζ) × D(ζ). Suppose that (5.3.4) holds. Then, ζ is closed if and only if Eq is complete.

62

Toka Diagana

Proof. Suppose that ζ is closed and let (us )s∈N ∈ D(ζ) be a Cauchy sequence in Eq . Thus kut − us k2q = max(|q(ut − us )|, kut − us k2 ) 7→ 0 as t, s → 7 ∞. Now from the fact that k · k2q ≥ |q(·)|, k · k2 it easily follows that kut − us k, |q(ut − us )| 7→ 0 as t, s 7→ ∞, and hence there exists u ∈ Eω such that us 7→ u as s 7→ ∞ and q(ut −us ) 7→ 0 in K as t, s 7→ ∞. In other words, (us )s∈N is ζ-convergent to u. Since ζ is closed it follows that u ∈ D(ζ) and q(us − u) 7→ 0 in K as s 7→ ∞. In summary, kus − ukq 7→ 0 as s 7→ ∞, and therefore, Eq is complete. Conversely, suppose that Eq is complete and let u = ζ − lim us for some sequence s→∞

(us )s∈N . Thus (us )s∈N ∈ D(ζ), us 7→ u in Eω , and q(ut − us ) 7→ 0 as t, s 7→ ∞. It is routine to see that kut − us kq 7→ 0 as t, s 7→ ∞. Now since Eq is complete there exists v ∈ D(ζ) = Eq such that kus − vkq 7→ 0 as s 7→ ∞. Now kus − vkq ≥ kus − vk, hence us 7→ v in Eω as s 7→ ∞. Using the uniqueness of the limit we deduce that u = v. We then conclude that q(us −u) 7→ 0 as s 7→ ∞, that is, q is closed. Remark 21. In view of the proof of Theorem 17, if ζ is a non-degenerate form whose corresponding quadratic form satisfies (5.3.4), then ζ − lim us = u if and only if kut − us kq 7→ 0 s→∞

as t, s 7→ ∞. Furthermore, u ∈ D(ζ), ζ − lim us = u if and only if kus − ukq 7→ 0 as s 7→ ∞. s→∞

If the non-degenerate form ζ is closed and if (5.3.4) holds, then the space Eq is called a p-adic Hilbert space when it is equipped with the K-form given by (5.3.2). Furthermore, if one supposes that ζ is bounded with bound ≤ 1 it is routine to check that the CauchySchwarz inequality is satisfied. Indeed, |hu, viζ | = |hu, vi + ζ(u, v)| ≤ max(|hu, vi|, |ζ(u, v)|) ≤ max(kukkvk, kukkvk) = kukkvk ≤ kukq kvkq for all u, v ∈ Eq . 5.3.3 Further Properties of the Closure Let φ, ψ be non-Archimedean bilinear forms. One says that φ is an extension of ψ and denote it ψ ⊂ φ if D(ψ) ⊂ D(φ) and ψ(u, v) = φ(u, v) for all u, v ∈ D(ψ). Definition 36. A non-Archimedean bilinear form φ is said to be closable if it has a closed extension. Theorem 18. Let φ be a non-degenerate non-Archimedean bilinear form. Suppose that q, the corresponding quadratic form to φ satisfies Eq. (5.3.4). If φ is closable, then φ − lim us = 0 yields q(us ) → 0 as s 7→ ∞. s→∞

Non-Archimedean Bilinear Forms

63

Proof. Let φ denote a closed extension of φ. Clearly, the statement φ − lim us = 0 yields s→∞

φ − lim us = 0 . Now if q is the quadratic form associated with φ it is clear that q(u) = q(u) s→∞

for each u ∈ D(φ) ⊂ D(φ), and hence q(us ) = q(us ) = q(us − 0) 7→ 0 as s 7→ ∞, by the closedness of φ. Remark 22. When φ is closable, the smallest closure in the sense of the extension of quadratic forms is called the closure of φ and denoted by φ. Clearly, D(φ) ⊂ D(φ).

5.4 Representation of Bilinear Forms on Eω × Eω by Linear Operators Definition 37. A bilinear form φ : D(φ) × D(φ) 7→ K is said to be representable whether there exists a (possibly unbounded) linear operator A : D(A) 7→ Eω such that φ(u, v) = hAu, vi, ∀u ∈ D(A), v ∈ D(φ). In this section we examine the representation of (unbounded) bilinear forms on Eω × Eω by linear operators. Namely, it will be shown that each non-degenerate bilinear form φ on Eω × Eω is representable whenever |φ(ei , e j )| lim = 0. (5.4.1) i→∞ kei k Moreover, if A denotes the linear operator on Eω associated with the form φ, then the adjoint A∗ of A does exist provided that the following holds: for each j ∈ N,

lim

i→∞

|φ(ei , e j )| kei k

= lim

i→∞

|φ(e j , ei )| kei k

= 0.

(5.4.2)

Theorem 19. Let φ : D(φ)×D(φ) 7→ K be a non-degenerate unbounded bilinear form. Then φ is representable whenever assumption (5.4.1) holds. Moreover, if A denotes the linear operator associated with φ, then the adjoint A∗ of A exists whenever assumption (5.4.2) holds. Proof. Suppose that assumption (5.4.1) holds. For all u = (ui )i∈N , v = (v j ) j∈N ∈ D(φ), write φ(u, v) =

∞

∑

φ(ei , e j ) ui v j , with ∀ j ∈ N, lim |ui | . |φ(ei , e j )|1/2 = 0. i→∞

i, j=0

Define the linear operator A on Eω as follows:     

D(A) := {u = (ui )i∈N ∈ Eω : lim |ui | kAei k = 0},

    Au =

i→∞

∑

i, j∈N

φ(ei , e j ) (e0j ⊗ ei )u, ∀u = (ui )i∈N ∈ D(A). ωi

64

Toka Diagana Obviously, A is well-defined. Indeed, for all j ∈ N, φ(ei , e j ) kei k = lim |φ(ei , e j )| = 0, lim i→∞ i→∞ ωi kei k

by using (5.4.1). Now 1 Au = ∑ j∈N ω j

!

∑ ui φ(ei , e j )

e j , ∀u = (ui )i∈N ∈ D(A),

i∈N

and hence for each i ∈ N, hAei , ei i = φ(ei , ei ). Moreover, D(A) ⊂ D(φ). Indeed, if u = (ui )i∈N ∈ D(A), then using the Cauchy-Schwarz inequality it follows that, ∀i ∈ N, |φ(ei , ei )| 2 2 |ui | |φ(ei , ei )| = |ui | . kei k . kei k |hAei , ei i| = |ui |2 . kei k . kei k ≤ |ui |2 . kei k . kAei k = (|ui | . kei k) . (|ui | . kAei k),

and hence lim |ui | . |φ(ei , ei )|1/2 = 0, that is, u ∈ D(φ). i→∞

Note that ui vk φ(ei , ek ) → 0 as i, k → ∞, by using the fact that (u ∈ D(A) ⊂ D(φ) and v ∈ D(φ)): |ui vk φ(ei , ek )| = |ui ||φ(ei , ek )|1/2 . |φ(ei , ek )|1/2 |vk | → 0, as i, k → ∞, and hence

∑ ∑ ui vk φ(ei , ek ) = ∑ ∑ ui vk φ(ei , ek ),

k∈N i∈N

i∈N k∈N

according to a result by Cassels [7]. Consequently, the following successive equalities are justified: ! 1 hAu, vi = ∑ ωk vk ∑ ui φ(ei , ek ) ωk i∈N k∈N ! =

∑ vk ∑ ui φ(ei , ek )

k∈N

=

∑

i∈N

φ(ei , ek )ui vk

i,k∈N

= φ(u, v) for all u = (ui )i∈N ∈ D(A) and v = (vk )k∈N ∈ D(φ). Furthermore, the uniqueness of A is guaranteed by the fact that φ is non-degenerate. It remains to show that A∗ , the adjoint of A exists; this can be done as in the bounded case.

Non-Archimedean Bilinear Forms

65

Example 27. Consider the bilinear form defined by φ(u, v) =

∑

πi j . ui v j , ∀u = (ui )i∈N , v = (vi )i∈N ∈ D(φ)

i, j∈N

where (πi j )i, j∈N ⊂ K is an arbitrary sequence, and the domain D(φ) of φ is defined as follows: D(φ) = {u = (ui )i∈N ∈ Eω : lim |ui | . |πi j |1/2 = 0}. i→∞

Note that φ(ei , e j ) = πi j for all i, j ∈ N and hence an equivalent of assumption (5.4.1) is: |πi j | = 0. i→∞ kei k lim

(5.4.3)

Upon making assumption (5.4.3), the unique (possibly unbounded) linear operator associated with φ is given by ! πi j 0 e j ⊗ ei u, ∀u = (ui )i∈N ∈ D(A) Au = ∑ i, j∈N ωi where D(A) = {u = (ui )i∈N ∈ Eω : lim kAei k . |ui | = 0}. i→∞ |πi j | |π ji | = lim = 0, then the adjoint A∗ of A does exist. If in addition, lim i→∞ kei k i→∞ kei k Corollary 6. Let φ : D(φ)×D(φ) 7→ K be a non-degenerate (unbounded) symmetric bilinear form. Then φ is representable whenever assumption (5.4.1) holds. Moreover, if A denotes the linear operator associated with φ, then the adjoint A∗ of A exists with A = A∗ .

5.5 Bibliographical Notes This chapter consists of a preliminary work by the author on non-Archimedean bilinear forms. All the results of this chapter are quite new and can be found in Diagana [11, 21].

Chapter 6

Functions of Some Self-adjoint Linear Operators on Eω 6.1 Introduction Fractional powers of closed linear operators play a key role in many fields such as the theory of analytic semigroups, existence and uniqueness of solutions of solutions to some partial differential equations, stochastic processes, and enable us to handle the well-known square root problem of Kato (see, [13], [14], [15], and [16]) 1 . Among other things, an important and challenging objective in a near future consists of formulating a non-Archimedean version of the square root problem of Kato. That obviously requires the introduction in the literature of a non-Archimedean theory of fractional powers for (general) closed unbounded linear operators. This chapter considers some preliminary investigations by the author on integer powers as well as functions of (possibly unbounded) of some specific linear operators on Eω . As an illustration, we will construct functions of a 2 × 2 symmetric matrix on Q p × Q p . However, one should point out that several questions related to fractional powers of general closed linear operators in the non-Archimedean context still remain.

6.2 Products and Sums of Diagonal Operators Let (λt )t∈N ⊂ K be a sequence satisfying (4.4.1) and let A be a diagonal operator defined on Eω by D(A) = {x = (xt )t∈N ∈ Eω : lim |λt | . |xt | . ket k = 0}, t

and Ax =

∑ λt xt et ,

∀x ∈ D(A).

t∈N

Now, let B ∈ U (Eω ) be another diagonal operator on Eω defined by, 1

Note that the square root problem of Kato in the classical setting amounts to showing that for some regular linear operators A, D(A1/2 ) = D(A∗1/2 ).

68

Toka Diagana D(B) = {x = (xt )t∈N ∈ Eω : lim |µt | . |xt | . ket k = 0}, t

and Bx =

∑ µt xt et ,

∀x ∈ D(B),

t∈N

where (µt )t∈N ⊂ K, then the algebraic sum A + B of A and B is defined by    D(A + B) = D(A) ∩ D(B),  

(A + B)x = Ax + Bx,

for all x ∈ D(A) ∩ D(B). As a straightforward consequence of Proposition 28 we have Corollary 7. Under (4.4.1), suppose that |µt | < |λt | for each t ∈ N, then A + B is selfadjoint. Furthermore, ρ(A + B) = {λ ∈ K : λ 6= λt + µt , ∀t ∈ N}, and 1 −1 k(A + B − λ) k = sup t∈N |λ − (λt + µt )| for each λ ∈ ρ(A + B). Proof. First of all, note that (A + B)x =

∑ (λt + µt )xt et for each x = (xt )t∈N ∈ D(A + B),

t∈N

where D(A + B) = {x = (xt )t∈N : lim |λt + µt | . |xt | . ket k = 0}. t→∞

From |µt | < |λt | for each t ∈ N it follows that |λt + µt | = |λt | for all t ∈ N, and hence, D(A + B) = D(A). Now, A + B is well-defined since lim |λt + µt | |xt | ket k = lim |λi | |xt | ket k = 0.

t→∞

t→∞

Note that A + B is diagonal with coefficients γt = λt + µt , where lim |γt | = lim |λt | = ∞, t→∞ t→∞ by (4.4.1). To complete the proof one follows along the same line as in the proof of Proposition 28. Let (µt ) ⊂ K be a sequence. Similarly, the product AB of the diagonal operators A and B is defined by:    D(AB) = {x ∈ D(B) : Bx ∈ D(A)},  

(AB)x = A(Bx), ∀x ∈ D(AB).

It can be easily checked that, (AB)x =

∑ λt µt xt et , for each x = (xt )t∈N ∈ D(AB), where

t∈N

D(AB) = {x = (xt )t∈N : lim |λt | . |µt | . |xt | . ket k = 0}. t→∞

Functions of Some Self-adjoint Linear Operators on Eω

69

Corollary 8. If lim |λt µt | = ∞, then the product AB of A and B is self-adjoint. Furthermore, t→∞

ρ(AB) = {λ ∈ K : λ 6= λt µt , ∀t ∈ N}, and 1 k(AB − λ)−1 k = sup t∈N |λt µt − λ| for each λ ∈ ρ(AB).

6.3 Integer Powers of Diagonal Operators For µ = (µt )t∈N ⊂ K, let J(µ) denote the collection of z ∈ Z such that lim |µt z | = lim |µt |z = ∞, that is,

t→∞

t→∞

J(µ) = {z ∈ Z : lim |µt z | = lim |µt |z = ∞}. t→∞

t→∞

Remark 23. If λ = (λt )t∈N , µ = (µt )t∈N are sequences of elements in K, one has the following properties: (1) If z, z0 ∈ J(λ), then z + z0 , zz0 ∈ J(λ); (2) J(λ) = J(|λ|); (3) J(λ + µ) = J(max(|λ|, µ|)) whenever |λ| 6= |µ|; (4) J(µ) ⊂ J(λ) whenever |µ| ≤ |λ|; (5) 0 6∈ J(λ) for each λ; (6) J(0) = Z− − {0}, the set of nonzero negative integers; / (7) J(λ) ∩ J(λ−1 ) = {0}; / (8) J(1K ) = {0}. Definition 38. Let z ∈ J(λ). Define integer powers (Az )z∈Z−{0} of the diagonal operator A by:  D(Az ) = {x = (xt )t∈N ⊂ K : lim |λt |z . |xt | . ket k = 0},   t  (6.3.1)  z z z   A x = ∑ λt xt et for each x = (xt )t∈N ∈ D(A ), t∈N

where (λt )t∈N ⊂ K. Similarly, one defines A0 = I, where I is the identity operator on Eω . Example 28. Suppose K = Q p where p ≥ 2 is a prime number. Consider the diagonal operator A defined by: D(A) = {x = (xt )t∈N ⊂ Q p : lim |λt | . |xt | . ket k = 0}, t

and

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Toka Diagana Ax =

∑ λt xt et ,

∀x ∈ D(A),

t∈N t

where λt = p p for each t ∈ N. It is easy to see that J(λ) = Z− − {0}, the set of all nonzero negative integers. In this event, for each z ∈ Z− − {0}, one defines Az by: t

D(Az ) = {x = (xt )t∈N ⊂ Q p : lim p−zp |xi |ket k = 0} t

and Az x =

∑ pzp xt et , t

∀x ∈ D(A).

t∈N

Using previous results, one can easily see that Az is self-adjoint and that t

ρ(Az ) = {λ ∈ Q p : λ 6= p p , ∀t ∈ N}. Proposition 31. Let z, z0 ∈ J(λ). If A is a diagonal operator with (λt )t∈N ⊂ K as a corresponding sequence, then 0

0

(1) Az .Az = Az+z ; 0 0 (2) (Az )z = Azz .

∑ xt et ∈ D(Az .Az ), then Az Az x = ∑ λt z+z xt et . Next we use the fact that 0

Proof. (1) If x =

0

0

t∈N

t∈N

z, z0 ∈ J(λ) yield z + z0 ∈ J(λ) (see Remark 29). 0 0 (2) Similarly, if x = ∑ xt et ∈ D((Az )z ), then (Az )z x = t∈N z z0

zz0

∑ λt zz xt et . Using Remark 29 it 0

t∈N

easily follows that (A ) = A . Proposition 32. If z ∈ J(λ), then Az is self-adjoint. Furthermore, ρ(Az ) = {λ ∈ K : λ 6= λtz , ∀t ∈ N}, and 1 z −1 k(A − λ) k = sup z t∈N |λt − λ| for each λ ∈ ρ(Az ). Proof. First of all, note that Az x =

∑ λt zxt et , ∀x = (xt )t∈N ∈ D(Az) with D(Az) = {x =

t∈N

(xt )t∈N ⊂ K : lim |λt |z |xt | ket k = 0}. Since z ∈ J(λ) it follows that Az is well-defined. t→∞

Note that Az is a diagonal operator corresponding to γt = λt z with lim |γt | = ∞, by t→∞

z ∈ J(λ). So to complete the proof one follows along the same line as in the proof of Proposition 28. Proposition 33. Let A, B be diagonal operators on Eω . If λ = (λt )t∈N , µ = (µt )t∈N are respectively the corresponding sequences to the diagonal operators A and B, and if |λt | 6= |µt | for each t ∈ N and J(λ) ∩ J(µ) ∩ Z+ 6= 0/ (Z+ being the set of all natural numbers), then D((A + B)z ) = D(Az ) ∩ D(Bz) = D((A + B)∗z ), for each z ∈ J(λ) ∩ J(µ) ∩ Z+ .

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Proof. Using the fact that |λt | 6= |µt | for each t ∈ N it easily follows that |λt + µt |z = max(|λt |z , |µt |z )

(6.3.2)

for all t ∈ N, z ∈ Z+ . In particular, (6.3.2) holds for each z ∈ J(λ) ∩ J(µ) ∩ Z+ . Now the operator (A + B)z is defined by: (A + B)z x =

∑ (λt + µt )z xt et

t∈N

for each x = (xt )t∈N ∈ D((A + B)z), where D((A + B)z) = {x = (xt )t∈N ⊂ K : lim |λt + µt |z |xt | ket k = 0} t→∞

= {x = (xt )t∈N : lim max(|λt |z , |µt |z ) |xt | ket k = 0} t→∞

z

z

= D(A ) ∩ D(B ). It is also clear that (A + B)z is self-adjoint for each z ∈ J(λ) ∩ J(µ) ∩ Z+ , and hence (A + B)z = (A + B)∗z. Remark 24. In view of Definition 38, the point now is how to define integer powers of a general unbounded linear operator A defined by Aψ = ∑ ats (e0s ⊗ et )ψ, ∀ψ ∈ D(A). ts

It is clear that whether A is a self-adjoint linear operator on Eω whose point spectrum is given by a sequence (γt )t∈N and if the sequence ( ft )t∈N is its corresponding orthogonal eigenfunctions, then A can be seen as a diagonal operator in ( ft )t∈N , and hence Definition 38 can be applied to it. More generally, if f : K 7→ K is an analytic functions whose domain Dom( f ) contains the sequence (γt )t∈N , then one can define f (A). The latter point will be investigated in Section 6.4. Another interesting question concerns fractional powers of a general unbounded linear operator A defined by Aφ = ∑ ats (e0s ⊗ et )φ, ∀φ ∈ D(A), ts

including diagonal operators. A partial answer to this question is given as follows: consider M , the set of all (complete) non-Archimedean fields (K, |·|) such that if λ ∈ K, then λq ∈ K for some q ∈ Q, and |λq | = |λ|q . Clearly, if K ⊂ M , then the fractional powers of every diagonal operator on (as a non-Archimedean Hilbert space over K) Eω are defined as for diagonal operators, see Definition 38. Next, one applies the previous method to those selfadjoint operators on Eω whose point spectrums are given by sequences as above to define their functions.

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6.4 Functions of Self-Adjoint Operators This section is a generalization of the previous one and introduces functions of some specific self-adjoint operators on Eω . Namely, we construct functions of a self-adjoint (possibly unbounded) operator T on Eω whose point spectrum σ p (T ), is given by a sequence (γt )t∈N ⊂ K. For that, let (ht )t∈N denote the sequence of eigenfunctions associated with the eigenvalues (γt )t∈N , that is, T ht = γt ht for each t ∈ N. As in the classical context, one can easily check that hht , hs i = ϖt δts for some (ϖt )t∈N ⊂ K−{0}. For the sake of simplicity, we suppose that (ht )t∈N is an orthogonal base for Eω . To achieve the latter goal, we assume that there exists a nontrivial isometric linear bijection V such that: Vet = ht for all t ∈ N. Consequently, hht , hs i = hVet ,Ves i = ϖt δts , and kht k2 = |ϖt | = |ωt | for each t ∈ N. Clearly, for each u ∈ Eω , u =

lim |ut |kht k = 0, where kht k = |ωi |1/2 , for ∑ ut ht with t→∞

t∈N

each t ∈ N. Moreover, the operator T can be rewritten in the base (ht )i∈N as  D(T ) = {u = (ut )t∈N ∈ Eω : lim |ut ||γt | kht k = 0},   t→∞     Tu :=

∑ γt ut ht ,

for each u = (ut )t∈N ∈ D(T ).

t∈N

Thus if f : K 7→ K is analytic, and if each γt for t ∈ N belongs to Dom( f ), the domain of f , one then defines f (T ) by  D( f (T )) = {u = (ut )t∈N ∈ Eω : lim |ut || f (γt )| kht k = 0},   t→∞     f (T )u :=

∑ f (γt )ut ht ,

for each u = (ut )t∈N ∈ D( f (T )).

t∈N

Using Proposition 28, the following holds: Proposition 34. Under previous assumptions, the operator f (T ) defined above is selfadjoint. Furthermore, ρ( f (T )) = {γ ∈ K : λ 6= f (γt ), ∀t ∈ N}, and 1 −1 k( f (T ) − λ) k = sup t∈N | f (γt ) − λ| for each λ ∈ ρ( f (T )).

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73

If f , g : K 7→ K are analytic functions such that σ p (T ) ⊂ Dom( f ) ∩ Dom(g), then the following hold: (1) (2) (3) (4)

(α f )(T ) = α f (T ) for all α ∈ K; D ( f (T ) + g(T )) ⊂ D (( f + g)(T )); D( f (T )g(T )) ⊂ D(( f g)(T )); If f (x) = at xt + at−1 xt−1 + ... + a1 x + a0 ∈ K[x], then f (T ) = at T t + at−1 T t−1 + ... + a1 T + a0 I

with D( f (T )) = D(T t ); (5) k f (T )k = sup | f (γt )|. t∈N

6.5 Functions of Some Symmetric Square Matrices Over Q p × Q p Throughout of this section, we let p denote an odd prime. This section illustrates in some extent our previous construction. Namely, if f : Q p 7→ Q p is an analytic function, we give an explicit formula for f (T ) where T is a 2 × 2 symmetric matrix over Q p × Q p and σ p (T ) ⊂ Q p . In particular, if f is either f (x) = xn or f (x) = x−n , or f (x) = exp p (x) for x ∈ Z p such −p

that |x| < p p−1 , then the formula for f (T ) will be considered. Here, we consider 2 × 2 (symmetric) square matrices T over Q p × Q p : ! a b T= , a, b ∈ Q p . b a From now on, we suppose that a, b ∈ Q p − {0}. Moreover, the direct sum Q p × Q p will be equipped!with the ! non-Archimedean norm and the inner product defined respectively by: x u For all , ∈ Q p × Q p, y v k

x y

and x h y

!

!

,

k = max(|x|, |y|) ! u i = xu + yv v

where | · | is the p-adic absolute value.! 1 In view of the above, ke1 k = k k = 1, ke2 k = k 0 hes , et i = δst , s,t = 1, 2, where δst are the Kronecker symbols.

0 1

!

k = 1, and

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It is ! not hard to see!that σ(T ) = σP (T ) = {a − b, a + b}. Moreover, it is clear that φ = 1 1 and ψ = are respectively the eigenvectors associated with the eigenvalues −1 1 a − b and a + b. Consider the nontrivial linear bijection V defined on Q p × Q p by   1 1   V=   −1 1 whose inverse V −1 is given by 

 V −1 = 

1/2

−1/2

1/2

1/2



 .

Moreover, Ve1 = φ ! and Ve2 = ψ. Clearly, {φ, ψ} is also an orthogonal base of Q p × Q p. x Furthermore, each = xe1 + ye2 ∈ Q p × Q p can be uniquely expressed as y x y

!

x+y x−y .φ+ . ψ. = 2 2

Clearly, T considered in {φ, ψ} can be rewritten as ! a−b 0 T= , a, b ∈ Q p , 0 a+b or x y

T

!

x+y x−y . φ + (a + b) . ψ. = (a − b) 2 2

Using our previous setting, if f : Q p 7→ Q p is an analytic function whose domain contains σ(T ), then

f (T )

x y

!

x+y x−y f (a − b)φ + f (a + b)ψ = 2 2 ! x+y x−y 1 f (a − b) + f (a + b) = 2 2 −1   x−y x+y f (a − b) + f (a + b)   2 2   =  x+y x−y f (a − b) + f (a + b) − 2 2

1 1

!

Functions of Some Self-adjoint Linear Operators on Eω

75

!

x ∈ Q p × Q p. y Clearly,

for all

 f (a + b) − f (a − b) f (a − b) + f (a + b)   2 2   f (T ) =  .  f (a + b) − f (a − b) f (a − b) + f (a + b)  2 2 Note that f (T ) is also a symmetric matrix whose eigenvalues are given by: 

σ( f (T )) = { f (a − b), f (a + b)} = f (σ(T )). 6.5.1 The Powers of the Matrix T We are interested to finding explicit expressions of both T n and T −n for n = 1, 2, 3, .... For that we take respectively f (x) = xn , x ∈ Q p and f (x) = x−n , x ∈ Q p − {0}. We have 

(a − b)n + (a + b)n  2  Tn =   (a + b)n − (a − b)n 2 If a − b, a + b ∈ Q p − {0}, then 

(a − b)−n + (a + b)−n  2  T −n =   (a + b)−n − (a − b)−n 2

 (a + b)n − (a − b)n  2  . n n (a − b) + (a + b)  2  (a + b)−n − (a − b)−n  2  .  (a − b)−n + (a + b)−n 2

6.5.2 Exponential of the Matrix T To define eT , the exponential matrix of T , we suppose that both a − b and a + b belong to −p the domain of the p-adic exponential, i.e., a + b, a − b ∈ Z p such that |a − b|, |a + b| < p p−1 . Under previous assumptions, eT is defined by:  exp (a − b) + exp (a + b) exp (a + b) − exp (a − b)  p p p p   2 2   eT =  ,  exp (a + b) − exp (a − b) exp (a − b) + exp (a + b)  p p p p 2 2 where exp p (·) denotes the p-adic exponential function. As previously, eT is also symmetric with eigenvalues given by: σ(eT ) = {exp p (a − b), exp p (a + b)} = exp p (σ(T )).

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6.6 Open Problems Problem 1. Let T be an arbitrary (possibly unbounded) self-adjoint linear operator on Eω . Construct fractional powers of T . Problem 2. Let T be an arbitrary (possibly unbounded) normal linear operator on Eω . As in Problem 1, Can we construct fractional powers of T ? Problem 3. Formulate a non-Archimedean version of the Kato’s square root problem for normal operators. For more on the classical version of the square root problem of Kato for (unbounded) normal operators, we refer the reader to Diagana[19]. Problem 4. Let A, B be (unbounded) self-adjoint operators on Eω such that A − B ∈ B2 (Eω ). Suppose that both f (A) and f (B) exist, where f : K 7→ K is an analytic function. Find (necessary) sufficient conditions under which f (A) − f (B) ∈ B2 (Eω ). In this event, approximate Q ( f (A) − f (B)) in terms of Q (A − B) (Q (T ) being the Hilbert-Schmidt norm of T ∈ B2 (Eω )). Problem 5. Let A, B be (unbounded) self-adjoint operators on Eω such that A − B ∈ B1 (Eω ). Suppose that both f (A) and f (B) exist for some analytic function f : K 7→ K. Find sufficient conditions under which f (A) − f (B) ∈ B1 (Eω ). In this event, find a relationship between tr( f (A) − f (B)) in terms of tr(A − B) (tr(T ) being the trace of T ∈ B1 (Eω )).

6.7 Bibliographical Notes The results of this chapter are quite new and entirely based upon results in Diagana[22].

Chapter 7

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces 7.1 Introduction Let (K, +, ·, | · |) be a (complete) non-Archimedean valued field and let Ωr be the closed ball of K centered at 0 with radius r, that is, Ωr = {κ ∈ K : |κ| ≤ r}. As we have previously seen, Ωr is also open in K. Furthermore, every ball Ωr is an additive subgroup of K. From now on, the radius r of the ball Ωr will be suitably chosen so that the series, which defines the p-adic exponential converges. Indeed, let K = Q p be the field of p-adic numbers (p ≥ 2 being a prime) equipped with the usual p-adic valuation | · | and let Ωr = {q ∈ Q p : |q| ≤ r}. As we have seen in (1.3.4) of Subsection 1.1.3., the p-adic exponential xt exp p (x) := ∑ t≥0 t! is not always well-defined and analytic for each x ∈ Q p . However, it does converge for all 1

x ∈ Z p such that |x| < r = p− p−1 . For more on these we refer the reader to Subsection 1.1.3 or [1, 40, 66]. In this chapter we provide the reader with a brief conceptualization of a nonArchimedean counterpart of the classical C0 -semigroups in connection with the formalism of linear operators on free Banach and non-Archimedean Hilbert spaces. The present chapter is mainly motivated by the solvability of p-adic differential and partial differential equations, as strong (mild) solutions to the Cauchy problem related to several classes of differential and partial differential equations arising in the classical context can be explicitly expressed through C0 -semigroups, see, e.g., [57] and [58]. As for the p-adic exponential defined above, here, the parameter of a given C0 semigroup belongs to one of those clopen balls Ωr whose radius r will be suitably chosen. Let us mention however that the idea of considering one-parameter families of bounded linear operators on balls such as Ωr was first initiated in [1] for bounded symmetric operators

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on Q p . Here, it goes back to consider those issues within the framework of free Banach and non-Archimedean Hilbert spaces while a development of a non-Archimedean operator theory is underway. One of the consequences of the ongoing discussion is that if K = Q p and if A is a 1

bounded linear operator on a free Banach space E satisfying kAk ≤ r with r = p− p−1 , then the function defined by ! (xA)t v(x) = ∑ u0 , x ∈ Ωr t≥0 t! for a fixed u0 ∈ E, is the solution to the homogeneous p-adic differential equation given by   d u(t) = Au(t), t ∈ Ωr dt  u(0) = u0

7.2 Basic Definitions Let (E, k · k) be a free Banach space. Throughout the rest of this chapter, we consider families (T (κ))κ∈Ωr : E 7→ E of bounded linear operators. We always suppose that r is suitably chosen so that κ 7→ T (κ) is well-defined. Definition 39. Let r > 0 be a real number. A family (T (κ))κ∈Ωr : E 7→ E of bounded linear operators will be called a semigroup of bounded linear operators on E if (1) T (0) = IE , where IE is the unit operator of E; (2) T (κ + κ0 ) = T (κ)T (κ0 ) for all κ, κ0 ∈ Ωr . The semigroup (T (κ))κ∈Ωr will be called of class C0 or strongly continuous if the following additional condition holds (3) lim kT (κ)x − xk = 0 for each x ∈ E. κ→0

Remark 25. One should point out that a semigroup (T (κ))κ∈Ωr of bounded linear operators is not only a semigroup but also a group. Every T (κ) is invertible, the inverse being T (−κ), according to (1)-(2) of Definition 39. Moreover, it is an infinite abelian group. Thus throughout the rest of this chapter the expression “group” (respectively, C0 -group) will be preferred to that of “semigroup” (respectively, C0 -semigroup). Remark 26. A group (T (κ))κ∈Ωr will be called uniformly continuous if the following additional condition holds (4) lim kT (κ) − IE k = 0. κ→0

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces

79

Definition 40. If (T (κ))κ∈Ωr is a group as above, then the linear operator A defined by  T (κ)x − x   exists}, D(A) = {x ∈ E : lim   κ→0 κ    T (κ)x − x    Ax = lim , for each x ∈ D(A) κ→0 κ

is called the infinitesimal generator associated with the group (T (κ))κ∈Ωr . Remark 27. (1) Note that if (T (κ))κ∈Ωr is a group on E and if (et )t∈N denotes the orthogonal basis for E, then T (κ) for each κ ∈ Ωr can be expressed as, ∀x = ∑ xt et ∈ E, by t∈N

T (κ)x =

∑ xt T (κ)et ,

t∈N

where ∀s ∈ N, T (κ)es =

∑ ats (κ)et

with lim |ats (κ)|ket k = 0.

t∈N

t→∞

(2) Using (1) one can easily see that for each 0 6= κ ∈ Ωr , ass (κ) − 1 ats (κ) T (κ) − IE ∀s ∈ N, es = es + ∑ et κ κ κ t6=s with lim |ats (κ)|ket k = 0. t6=s,t→∞

(3) If (T (κ))κ∈Ωr is a group on E, then its infinitesimal generator A may or may not be a bounded linear operator on E.

7.3 Properties of non-Archimedean C0 -Groups In this chapter we mainly focus on general groups, and strongly continuous groups of bounded linear operators on general free Banach spaces. We begin with the following example: Example 29. Take K = Q p the field of p-adic numbers. Consider the ball Ωr of Q p with 1

r = p− p−1 . Let (X, k · k) be a free Banach space over Q p and let ( ft )t∈N be its canonical orthogonal base. Define for each q ∈ Ωr and for x = ∑ xt ft ∈ X the family of linear t≥0

operators

T (q)x = ∑ xt eµt q ft t≥0

where (µt )t∈N ⊂ Ωr is a sequence of nonzero elements. It is routine to check that the family (T (q))q∈Ωr is well-defined.

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Proposition 35. The family (T (q))q∈Ωr of linear operators given above is a C0 -group of bounded linear operators whose infinitesimal generator is the (bounded) diagonal operator A defined by Ax = ∑ µt xt ft for each x = ∑ xt ft ∈ X. t≥0

t≥0

Proof. First, note that T (q) is analytic on the ball Ωr . It is routine to check that (T (q))q∈Ωr is a family of bounded linear operators on X. Indeed, for each q ∈ Ωr , ! s qs µ ft , ∀t ∈ N, T (q) ft = eµt q ft = ∑ t s≥0 s! and hence

||T (q)|| =

µts qs ∑ s≥0 s!

by the fact that qµt ∈ Ωr for each t ∈ N. Furthermore, one can easily check that

! < ∞,

(1) T (0) = IX ; (2) T (q + q0 ) = T (q)T (q0 ) for all q, q0 ∈ Ωr ; (3) lim kT (q)x − xk = 0 for each x ∈ X. q→0

Thus (T (q))q∈Ωr is a C0 -group of bounded linear operators. Now let B be the infinitesimal generator of (T (q))q∈Ωr . It remains to show that A = B. First of all, let us show that D(B) = X (= D(A)). Clearly, for each 0 6= q ∈ Ωr , µt q T (q) ft − ft e −1 = ft q q for each t ∈ N, and hence

T (q) ft − ft

= 0} = X, D(B) = {x = (xt )t∈N : lim |xt | .

t→∞ q by

for each x =

T (q) ft − ft (|xt |k ft k)

≤

7→ 0 as t 7→ ∞, |xt | .

q |q|

∑ xt ft ∈ X.

t∈N

To complete the proof it suffices to prove that

A ft − T (q) ft − ft 7→ 0 as q 7→ 0.

q µt q e −1 The latter is actually obvious since 7→ µt as q 7→ 0, and hence B = A is the q infinitesimal generator of the C0 -group (T (q))q∈Ωr .

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces

81

In the next theorem, we take K = Q p where p ≥ 2 is prime. Note also that it is a natural generalization of Example 29. 1

Theorem 20. Let A be a bounded linear operator on X such that kAk < r = p− p−1 . Then A is the infinitesimal generator of an uniformly continuous group of bounded operators (T (q))q∈Ωr . 1

Proof. Suppose that A is a bounded linear operator on X with kAk < r = p− p−1 and set, for each q ∈ Ωr , T (q) =

(qA)s . s≥0 s!

∑

(7.3.1)

Clearly, the series given by (7.3.1) converges in norm and defines a family of bounded linear operators on E, by |q| . kA|| < r. It is also routine to check that T (0) = IX , T (q+q0 ) = T (q)T (q0 ) for all q, q0 ∈ Ωr . It remains to show that (T (q))q∈Ωr given above is uniformly continuous. Indeed, 0 6= q ∈ Ωr , one has ) ( (qA)s , T (q) − IX = qA ∑ s≥0 (s + 1)! and hence

T (q) − IX

− A

≤ kAk . kT (q) − IX k < kT (q) − IX k . q Now, kT (q) − IX k ≤ |q| . kAk . kζ(q)k, where ζ(q) =

(7.3.2)

(qA)s

∑ (s + 1)! converges, and hence

s≥0

lim kT (q) − IX k = 0.

q7→0

Consequently,

(7.3.3)

T (q) − IX

− A lim

= 0, q7→0 q

by using both (7.3.2) and (7.3.3). Remark 28. (i) Note that the mapping Ωr 7→ B(Eω ), q 7→ T (q) is analytic. Furthermore, dT (q) = AT (q) = T (q)A. dt (ii) An abstract version of Theorem 20 , i.e., in a general non-Archimedean valued field K, remains an open problem. Now let (K, | · |) be a (complete) non-Archimedean valued field and let Ωr ⊂ K be a clopen, were r is chosen so that Ωr 7→ B(X), κ 7→ T (κ) is well-defined.

82

Toka Diagana

Theorem 21. Let (T (κ))κ∈Ωr be a C0 -group and let A be its infinitesimal generator. Suppose that there exists M > 0 such that kT (κ)k ≤ M for each κ ∈ Ωr ⊂ K. Then, for each x ∈ D(A), T (κ)x ∈ D(A) for each κ ∈ Ωr . Furthermore, dT (κ) x = AT (κ)x = T (κ)Ax. dκ Proof. The proof, in some extent, is similar to that of the classical one, however for the sake of clarity, we will provide the reader with all details. Let x ∈ D(A) and let 0 6= κ ∈ Ωr . Using Definition 39, Definition 40, and the boundedness of the C0 -group T (κ), it easily follows that T (κ) − IX T (κ) − IX 0 0 T (κ )x = T (κ ) x (7.3.4) κ κ (7.3.5) 7→ T (κ0 )Ax as κ 7→ 0. Consequently, T (κ0 )x ∈ D(A) and AT (κ0 )x = T (κ0 )Ax, by (7.3.4). Furthermore, since T (κ) − IX 0 x 7→ T (κ0 )Ax, as κ 7→ 0 T (κ ) κ it follows that the right derivative of T (κ0 )x is T (κ0 )Ax. To complete the proof, we have to show that for each 0 6= κ0 ∈ Ωr , the left derivative of T (κ0 )x exists and is T (κ0 )Ax. Note that if σ, σ0 ∈ Ωr , then so is σ − σ0 , by using |σ − σ0 | ≤ max(|σ|, |σ0 |) < r. Now T (κ0 )x − T (κ0 − κ)x 0 lim − T (κ )x = κ→0 κ

T (κ)x − x lim T (κ − κ) − Ax κ→0 κ + lim T (κ0 − κ)Ax − T (κ0 )Ax . 0

κ→0

Clearly,

T (κ)x − x − Ax = 0, lim T (κ − κ) κ→0 κ 0

by kT (σ)k ≤ M for each σ ∈ Ωr . Using the strong continuity of the group T (κ) it follows that lim T (κ0 − κ)Ax − T (κ0 )Ax = 0. κ→0

Consequently, lim

κ→0

T (κ0 )x − T (κ0 − κ)x 0 − T (κ )x = 0, κ

and hence the left derivative of T (κ0 )x exists and equals T (κ0 )Ax. This completes the proof.

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces

83

Remark 29. One of consequences of Theorem 21 is that the function v(t) = T (t)u0 , t ∈ Ωr for some u0 ∈ D(A), is the solution to the homogeneous p-adic differential equation given by  du   = Au(t), t ∈ Ωr ,  dt    u(0) = u , 0

(7.3.6)

where A : D(A) ⊂ X 7→ X is the infinitesimal generator of the C0 -group (T (t)t∈Ωr and u : Ωr 7→ D(A) is a X-valued function.

7.4 Existence of Solutions to Some p-adic Differential Equations The present section continues the above-mentioned discussion. Namely, we study the existence and uniqueness of a solution to the Cauchy problem for the homogeneous p-adic differential equation  du   = Au(t), t ∈ Z p ,  dt (7.4.1)    u(0) = u ∈ Q , 0 p where A is a self-adjoint linear operator on C(Z p , Q p ) whose spectrum σ(A) = σ p (A) = {λk }k∈N with σ p (A) being the point spectrum of A. For that, the first task consists of constructing the spectral decomposition of A. Then utilizing the spectral decomposition of A and upon making some additional assumptions on σ(A), it will be shown that (7.4.1) has a unique solution, which can be explicitly expressed as u(t) =

∑ etλ Ei(u0 ),

∀t ∈ U p ,

i

i∈N

where (Ei )i∈N is called spectral projection associated with A in the sense that Ei E j = E j Ei = 1

δi j Ei for all i, j ∈ N, and U p = {u ∈ Z p : |u| < p− p−1 }. However, one should point out that the question, which consists of the existence and uniqueness of a solution to the corresponding inhomogeneous equation to (7.4.1) is open. Furthermore, except results of [31, 32] on the existence of solutions to (7.4.1) in the case when A is closed in some non-Archimedean Banach space, it is quite unclear whether our results can be compared with the rest of the existing literature on p-adic differential equations. To discuss the existence of solutions to (7.4.1), we first establish a spectral decomposition for A, which enables us to construct the C0 -semigroup associated A. To do so, we require the following assumptions:

84

Toka Diagana

(H.1) The operator A is a self-adjoint on C(Z p , Q p ) with spectrum σ(A) consisting of its eigenvalues {λk }k∈N . Each eigenvalue is of (finite) multiplicity ml that is equal to the dimension of the corresponding eigenspace. (H.2) There exists an orthonormal base ( fkl )k∈N of eigenvectors for C(Z p , Q p ). 1

(H.3) σ(A) ⊂ U p where U p = {u ∈ Z p : |u| < p− p−1 }. Theorem 22. Under assumptions (H.1)-(H.2), for each u ∈ D(A), mk

Au =

∑ λk ∑ hu, fkl i fkl =

k∈N

l=1

∑ λk Ek (u)

k∈N

mk

where Ek (u) =

∑ hu, fkl i fkl for each k ∈ N, is called spectral projection associated with A

l=1

in the sense that Ei E j = E j Ei = δi j Ei for all i, j ∈ N. Proof. Using assumption (H.2) it easily follows that each u ∈ C(Z p , Q p ) can be expressed in the orthonormal base ( fkl )k∈N as follows:

mk

mk

u = ∑ ∑ hu, fkl i fkl = ∑ Ek (u) with lim ∑ hu, fkl i fkl = 0,

k→∞ k∈N l=1 k∈N l=1 ∞

mk

where Ek (u) = ∑ hu, fkl i fkl . l=1

Now mk

Au =

∑ ∑ hu, fkl iA fkl

k∈N l=1

mk

=

∑ λk ∑ hu, fkl i fkl

k∈N

=

l=1

∑ λk Ek (u)

k∈N

mk

provided that lim λk ∑ hu, fkl i fkl = lim kλk Ek (u)k∞ = 0, that is, u ∈ D(A).

k→∞ k→∞ l=1 ∞ Clearly, for each u ∈ C(Z p , Q p ), mi

Ei E j (u) =

∑ hE j (u), fil i fil

l=1 mi m j

=

∑ h ∑ hu, f jr i f jr , fil i fil

l=1 r=1 mi m j

=

∑ ∑ hu, f jr ih f jr , fil i fil ,

l=1 r=1

and hence Ei E j (u) = E j Ei (u) = δi j Ei (u) for all i, j ∈ N.

One-Parameter Family of Bounded Linear Operators on Free Banach Spaces

85

Now if f : Z p ∩ σ(A) 7→ Q p is analytic, one defines f (A) as follows:  lim | f (λi )| . kEi (u)k∞ = 0},   D( f (A)) := {u ∈ C(Z p , Q p ) : i→∞    

f (A)u =

∑ f (λi )Ei(u),

∀u ∈ D( f (A)).

i∈N

In particular, under assumption (H.3) and letting f (t) = et for each t ∈ U p it follows that for each t ∈ U p , etA u = ∑ eλi t Ei (u), u ∈ C(Z p , Q p ). i∈N

It is then easy to see that (etA )t∈U p is an analytic C0 -group, which turns out to be an infinite group satisfying etA = I for t = 0 with I being the identity operator of C(Z p , Q p ). Moreover, using the fact that AEi(u0 ) = λi Ei (u0 ) for each i ∈ N, one can easily see that u(t) =

∑ eλ t Ei (u0 ), i

t ∈ Up

i∈N

is the only solution to (7.4.1). The previous discussion can be formulated as follows: Theorem 23. Under assumptions (H.1)-(H.2)-(H.3), the first-order differential equation, (7.4.1), has a unique solution, which can be explicitly expressed as u(t) =

∑ eλ t Ei (u0 ), i

t ∈ Up,

i∈N mi

where Ei (u0 ) = ∑ hu0 , fil i fil for i ∈ N. l=1

Remark 30. For r > 0, let Ur = {u ∈ Q p : |u| ≤ r}. Using Theorem 23, one easily sees that the Cauchy problem for the p-adic partial differential equation  ∂u    (t, x) = Au(t, x), t ∈ Z p , x ∈ Ur ∂t (7.4.2)    u(0, x) = u0 (x), x ∈ Ur has also a unique solution, u : Z p × Ur 7→ Q p , which can be explicitly expressed as u(t, x) =

∑ eλ t Ei (u0 (x)), i

i∈N mi

where Ei (u0 (x)) = ∑ hu0 (x), fil i fil for i ∈ N. l=1

t ∈ U p,

86

Toka Diagana

7.5 Open Problems Problem 1. Consider the nonhomogeneous p-adic differential equation  du   = Au(t) + f (t), t ∈ Ωr ,  dt    u(0) = u ∈ Q , 0 p

(7.5.1)

where f : Ωr 7→ Q p is a continuous function. Express the solution to this equation in terms of u0 , f , and the C0 -group associated the multiplication operator Au = Q(t)u, ∀u ∈ C(Z p , Q p ), ∞

where Q =

∑ qs fs ∈ C(Z p, Q p ).

s=0

Problem 2. Prove the existence of solutions to the first-order p-adic differential equation  du   = Au(t) + f (t), t ∈ Z p ,  dt (7.5.2)    u(0) = u ∈ Q , 0

p

and the first-order p-adic partial differential equation  ∂u    (t, x) = Au(t, x) + f (t, x), t ∈ Z p , x ∈ Ωr ∂t    u(0, x) = u0 (x) ∈ C(Ωr , Q p ),

(7.5.3)

where A is a self-adjoint linear operator on C(Z p , Q p ) whose spectrum σ(A) = σ p (A) = {λk }k∈N with σ p (A) being the point spectrum of A and f ∈ C(Z p , Q p ) (respectively f : Z p × Ωr 7→ Q p is jointly continuous). Express the solution to this equation in terms of u0 , f , and the C0 -group associated the operator A.

7.6 Bibliographical Notes The results of this chapter are quite new and entirely based upon results in Diagana [23].

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Index φ-convergence, 58 p-adic absolute value, 6 p-adic differential equation, 77, 83 p-adic exponential, 9, 77 p-adic numbers, 1, 6 Eω , 33, 42, 46 adjoint operator, 27, 45 ball, 4, 77 Banach algebra, 22 base, 29 bounded operator, 19 Cauchy sequence, 3 Cauchy-Schwarz inequality, 62 clopen, 4, 5, 81 closable operator, 46 closure, 46, 58, 63 completely continuous operator, 39 convergence, 2 diagonal operator, 49, 69 domain of an operator, 43 eigenfunction, 24, 72 eigenvector, 24 field of formal Laurent series, 9 field of fractions, 4 form sum, 53, 59 free Banach space, 13, 14 function operator, 72 graph norm, 47 graph of an operator, 48 group, 79

Hensel decomposition, 7 Hilbert-Schmidt, 32, 38, 39 index metric, 3 integer powers, 67 inverse operator, 23 invertible operator, 23 matricial representation, 25 maximal ideal, 4 metric, 2 multiplication operator, 22 non-archimedean Banach space, 12 non-Archimedean bilinear form, 53 non-archimedean Hilbert space, 15 non-Archimedean quadratic form, 57 non-archimedean valued field, 2 one-to-one, 23 onto, 23 orthogonal base, 29 orthonormal base, 30 p-adic differential equation, 83 p-adic partial differential equation, 85 perturbation of bases, 29 pointwise convergence, 26 power series, 8 product of operators, 47, 68 quadratic form, 53 quadratically disjoint, 59 resolvent, 24 resolvent equation, 24 semigroup, 77, 78 spectrum, 24

92 sphere, 4 subring, 4, 7 sum of operators, 47, 68

Index trace, 40 unbounded operator, 43 unit ball, 4