On Some Classes Of Operators On Hilbert Space

Abstract

In this project, H will d8note a Hilbert space with inner projuct denoted by < ,7 and T, A, B, X etc. will denote operators (i.e. bounded, linear trans- formations) on a Hilbert space H into itself or into another Hilbert space.Ker T, ran T, 1 T, ran1 T

will denote the kernel of T, range of T, orthogonal ~' complement of ker T, orthogonal complement of ran T, respectively. B(H) will denote the Banach algebra of all operators on H. If A,BEB(H) then [A,B] will denote AB-BA ~ and FR will denote the fields of Gomplex and real numbers, respectively.

Introduction

The study of normal operators has been very suc- cessful in the sense that a lot of interesting results has been obtained concerning these operators e.g. the classical Putnam - Fuglede theorem, which will be stated later. Many authors have defined new classes

of operators by making them satisfy certain known properties of normal operators in the hope that some of the 'results which hold for normal operators, will also hold for these new classes of operators.

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For example, spectraloid operators have been defined using some properties of the sepectrum, spectal radius, etc. of normal operators. Others have defined other classes of operators by generalising the concept of normality e.g. binormal operators [2]. By relaxing the condition of normality we can also define other classes of operators e.g. dominant operators [17] and try to see which properties of normal operators will still hold for these larger classes of operators.

In this project, we have studied spectraloid, binormal and dominantoperators. For each of these three classes of operators, we have studied their properties, subclasses and the properties of these sub- classes. We have also looked at extensions of the Putnam-Fuglede theorem qnd finally we have given a problem which can lead to further research in this area of operator theory.

Preliminary defintions and results.

(i) The spectrum of an operator T, denoted by o(T), is defined by o(T) = {AE ~ : AI - T is not invertible} where I = identity operator.

(ii) The spectral radius of T, denoted by r(T), is defined by r(T) = sup {[A[:A E o(T)}.

(iii) The numerical range of an operator T, denoted by W (T), is defined by W (T) = {.(Tf,f)"":

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II f II = I}.

(iv) The numerical radius of an operator T, denoted by w(T), is defined by w(T) = sup {I A I : AE W(T)}

(v) A set A is said to be convex if for every x,y E A, tx +(l-t)y E A, where 0 < t < I

(vi) The convex hull of a set M denoted by conv M, is the intersection of all the convex sets which contain M. Note that conv M is itself convex since it is the intersection of convex sets and it is the smallest convex set which contains M.

(vii) Two operators A and B are said to be similar if there exists an invertible P such that p-l AP = B.

The following result-gives some properties of a normal operator ([7]).

Theorem A

true:

Let T be a normal operator. The following are

(a) r,.(T) = w(T)

(b) the'~losure of the numerical range of T is the, convex hull of its spectrum i.e W(T) = conv (o(T».

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(c) w(T) = II T II

Using this result, we can now define spectra­

loid operators.