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Operators Similar to Their Adjoints ––––––––––––––––––––––––––––— Popular and Democratic Republic of Algeria University of Oran 1 Ahmed Ben Bella Faculty of exact and applied Sciences Department of Mathematics Operators Similar To Their Adjoints THESIS SUBMITTED FOR THE DEGREE OF DOCTORATE IN MATHEMATICS Presented by : Dehimi Souheyb Doctorate’sThesis Committee : President: Professor C.Bouzar University of Oran 1 Ahmed Ben Bella Thesis Supervisor: Professor M.H.Mortad University of Oran 1 Ahmed Ben Bella Examiners: Professor K.Belghaba University of Oran 1 Ahmed Ben Bella Dr. M.Meftah (M.C.A) University of Oran 1 Ahmed Ben Bella Professor B.Benahmed National Polytechnic School of Oran (ENPO Ex: ENSET) Professor M.Tlemçani University of Sciences and Technology of Oran (M.B) Academic year 2016 2017 Acknowledgement I would like to thank first and for most my supervisor, Professor Mohammed Hichem Mortad, for his many suggestions, helpful discussions, patience and constant support during this research. I sincerely thank Professor C.Bouzar for giving me the honor of being president of the jury. I thank Professors K.Belghaba, B.Benahmed and M.Tlemçani and Doctor M.Meftah for their time and effort participating in my thesis committee. Of course, I am grateful to my parents for their support, encouragement, and patience. Without them this work would never have come into existence. Contents Introduction vi 1 Essential background 11 1.1 Banachalgebra ................................. 11 1.1.1 Introduction............................... 11 1.1.2 Basic properties of spectra . 12 1.2 C-algebra.................................... 13 1.3 Boundedoperators ............................... 15 1.3.1 Definitions and properties . 17 1.3.2 Approximate point spectrum . 18 1.3.3 Resolutions of the identity . 19 1.4 Polar decomposition of an operator . 21 1.4.1 Isometry and partial isometry . 22 1.4.2 Polar decomposition of an operator . 22 1.5 Positiveoperators................................ 25 1.6 Numericalrange................................. 27 2 Non-normal operator classes 29 2.1 Compactoperators ............................... 29 2.2 Hyponormal operators . 30 2.2.1 Definitions and properties . 31 i 2.2.2 Some conditions implying normality or self-adjointness . 33 2.2.3 p-Hyponormal operators . 35 2.3 Normaloid operators . 39 2.4 Paranormal operators . 42 2.4.1 Definitions and properties . 42 2.4.2 k-paranormal operators . 44 2.5 Convexoid operators . 46 2.5.1 Examples ................................ 47 2.6 ClassAoperators................................ 51 2.6.1 Quasi-class A operators . 54 3 Similarities involving bounded operators 56 3.1 Introduction................................... 56 3.2 Operators similar to their adjoints . 58 3.2.1 Conditions implying self-adjointness of operators . 58 3.2.2 Operators similar to self-adjoint ones . 63 3.3 Operators with inverses similar to their adjoints . 64 3.3.1 Operators similar to unitary ones . 64 3.3.2 Operators with left inverses similar to their adjoints . 69 3.4 Similarities involving normal operators . 73 3.5 Quasi-similarity of operators . 76 4 Similarities involving unbounded operators 78 4.1 Preliminaries .................................. 78 4.1.1 Adjoint Operators . 80 4.1.2 Self-adjoint Operators . 82 4.2 Quasi-similarity of unbounded operators . 86 4.3 Similarities involving unbounded normal operators . 90 ii 4.4 Unbounded operators similar to their adjoints . 94 Bibliography 99 iii Résumé Le but de ce travail est d’étudier la similarité entre les opérateurs linéaires et leurs adjoints dans un espace de Hilbert. Le travail présenté est organisé selon le plan suivant: Dans les deux premiers chapitres, un rappel sur des notions essentiel les sur les opérateurs linéaires bornés. Dans les deux derniers chapitres nous donnons quelques conséquences de la similarité et quasi-similarité entre les opérateurs linéaires. L’un des principaux objectifs de cette thèse est de généraliser le théorème de Sheth. Abstract The aim of this work is to study the similarity between linear operators and their adjoints in a Hilbert space. The work is organized according to the following plan. In the first two chapters, a reminder on essential notions about bounded linear operators. In the last two chapters we give some consequences of the similarity and quasi-similarity between the linear operators. One of the main objectives of this thesis is to generalize Sheth’stheorem. Keywords: similarity, bounded and unbounded operators, closed, self-adjoint, normal, hyponomal operators, unitary cramped operators, Sheth. iv Notations k The field of real or complex numbers. (T ) The spectrum of T. r(T ) The spectral radius of T. (H) The Banach algebra of all bounded linear operators. B ran(T ) The range of a linear operator T. T The adjoint operator of T. p (T ) The point spectrum of T. app (T ) The approximate point spectrum of T. PM Initial projection PN Final projection W (T ) The numerical range of T. (X, Y ) The set of all compact linear operators. B1 co (T ) The convex hull of (T ) . e (T ) The essential spectrum of T. D (T ) The domain of a linear operator T. (T ) The graph of an operator T. G T The closure of the closable operator T. v Introduction The theory of Banach algebras is an abstract mathematical theory which is the synthesis of many specific cases from different areas of mathematics. They are named after the Polish mathematician Stefan Banach (1892—1945) who had introduced the concept of Banach space, but Banach had never studied Banach algebras. In fact, the first one who have defined them is Mitio Nagumo in 1936 ([Nag36]) under the name “linear metric rings”. In 1941, I.M.Gelfand (1913—2009) introduced them under the name “normed rings”([Gel41]). In the classical monograph [BD73], the authors write that, if they had it their way, they would rather speak of “Gelfand algebras”. But in 1945, Warren Ambrose (1914—1995) came up with the name “Banach algebras”([Amb45]) . Banach algebras show up naturally in many areas of analysis: Let X be a Banach space. then (X) , the algebra of all bounded linear operators on X, is a Banach algebra, B with respect to the usual operator norm. If we have an analytic object that has a Banach algebra naturally associated with it, then this algebra can provide us with further insight into the nature of the underlying object. A notion which will be of great use in this thesis is that of hyponormal operator. A bounded hyponormal operator is a bounded operator T on a Hilbert space H such that T T TT . This definition was introduced by Paul Halmos [Halm50] in 1950 and ≥ generalizes the concept of a normal operator (where T T = TT ). The important thing is that there is a prominent example of a hyponormal operator, the unilateral shift. If l2 vi is the Hilbert space of square summable sequences and T is defined on l2 by T (x0, x1,...) = (0, x0, x1,...) . Then T is called the unilateral shift and is the most basic example of hyponormal operators. The unilateral shift is a well understood non normal operator; it is arguably the best understood non normal operator on an infinite dimensional space. Paul Halmos began a strategic attack on operator theory by extracting two properties of the shift in [Halm50]. One was the definition of hyponormal operators and the other the idea of a subnormal operator. A subnormal operator is one that has a normal extension; every subnormal operator is hyponormal. One of the first important results in the theory of hyponormal operators, due to C. R. Putnam, is the fact that if T is a pure hyponormal operator, then its real and imaginary parts, X and Y , must be absolutely continuous self- adjoint operators [Put63]. That is, the spectral measures for X and Y must be absolutely continuous with respect to Lebesgue measure on the real line. Thus the Spectral Theorem for self-adjoint operators can be applied to X, and this operator can be represented as a multiplication operator on L2 [a, b] for some interval in R. The operator Y also has such a representation, but on a different L2 space. Can Y be represented on the same space L2 [a, b] in a way that is intimately connected with the representation of X ? Indeed, a result of Kato [Kat68], though it is not directly related to hyponormal operators, implies that this can be done. About the same time, many authors began investigating hyponor- mal operators from this perspective [Pin68, Rus68, Xia63][Rus68][Xia63]. In addition if T is hyponormal, then T T TT Area ( (T )) . k k ≤ This was proved by Putnam in [Put70] and is well known as Putnam’sinequality. An operator T (H) is said to be p-hyponormal if 2 B p p (TT ) (T T ) ≤ vii for a positive number p. In fact, semi-hyponormal operators were first introduced by Professor D. Xia in [Xia80]. He also provides an example of a semi-hyponormal operator which is not hyponormal. In [Xia80] Xia proved that, if T is p-hyponormal, then 2p 1 T T TT p dd k k ≤ ZZ(T ) for p 1 . Ch¯oand Itoh proved that Putnam’sinequality holds for p hyponormal oper- ≥ 2 ators in [CI95] . The notion of a paranormal operator dates back to 1960s and is due to V. Istra¸tescu¼ in [Ist66] he named them “operators of class N ”. T. Furuta in [Fur67] introduced the term “paranormal operators”. The class of paranormal operators can be seen as a generalization of other important classes: hyponormal operators and subnormal and normal operators . In subsequent years paranormal operators have been the subject of further research. For example, we know that a paranormal operator T is compact if and only if T n is compact for some n N. Moreover, compact paranormal operator is normal . Paranormality 2 appears also to be an important property when studying various problems in operator theory.
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