## Maximal numerical range of Operators in hilbert spaces

##### Abstract

The maximal numerical range of a linear operator T, on a Hilbert space H,
denoted by Wo(T), is a subset of the complex field C. It is a relatively new
concept in ..operator theory, having been introduced only in 1970 by
Stampfli. It owes part of its motivation to the usual numerical range, WeT).
It is therefore important that the two concepts, that is, W(T) and WoCT)be
defined alongside one another so as to provide a good understanding of the
latter.
Definition
Let T be a bounded linear operator on a complex Hilbert space H. Then,
(1) the numerical range of T is defined to be the set; NA~qf1r
c.JlyOF 'B'
WeT) = {< Tx,x >: x E H, IIx II = 1 } dN~~;gtr'g L.I~H~J\
(2) the maximal numerical range of T is defined to be the set;
A lot of research has been done with the usual numerical range. Following is
an outline of some of the research works in this area:
Donoghue(1957) studied the numerial range of a bounded operator.
Stampt1i(1966) looked at the extreme points of the numerical range of a
hyponormal operator. Jointly with Williams, Stampfli(1968) went further
and studied the growth conditions and the numerical range in a Banach
algebra. Halmos( 1967) in -his book , " A Hilbert space problem book" ,
dedicated an entire chapter to the numerical range. A few years later,
Bonsall and Duncan(197l) wrote a book on numerical ranges of operators
on normed spaces and of elements of normed ·algebras. Embry(1971) used
subsets associated with. the numerical range to classify special operators. . " DashEi9.73) considered tensor products and joint numerical range.
Lancaster(1975) studied the boundaryof the numerical range, he introduced
the concept of essential numerical range and proved two results which
indicate a set theoretic relationship between the boundary of the numerical range and the essential numerical range. These were to be re-visited later by
Williams(1977), who tried to simplify the proofs. A study of the growth of
numerical ranges of powers of Hilbert space operators was done by
Shiu(l976). Khalagai(1979) in his M.Sc. thesis, did not only discuss the
topological properties of the numerical range, but he also showed the
bearing that the numerical range has got on properties of operators. He
showed how some conditions can be imposed on the numerical range of a
bounded operator in order to achieve normality, similarity, uniticity and self
adjointness of the operator. He further showed how the conclusion of
normality from classes of operators such as quasinormal, hyponormal,
paranormal and quasihyponormal can be attained via the numerical range.
Finally, he showed how the numerical range can be used to achieve the
positivity of a product of operators.
On the contrary, however, the maximal numerical range has not enjoyed as
much exposition. Stamp fli(1970), though using it as a powerful tool in
determining the norm of a derivation, seemed not to pay much interest to it,
but instead treated it casually. It is nevertheless remarkable to note that
Fong(l979) considered the essential maximal numerical range as an
independent subject. The work of Sheth and Duggal(l984) is also worth
being acknowledged. They initiated the study of maximal numerical range as
a subject in itself. Other major contributors in this field are Khan(l988), who
re-visited the idea of essential maximal numerical range; and Cho(l988),
whose result about the joint maximal numerical range is of particular
importance in this area.·70~ .
One can clearly see from the foregoing that the maximal numerical range is
still a virgin area of study as opposed to the usual numerical range. This
project is therefore geared towards providing a curtain-raiser for future
research in the area of maximal numerical range. To achieve this goal, the
project has been divided into four chapters.
In chapter one, the topological; properties of the maximal numerical range
are established: Towards this end, a comparative study of the usual
numerical range and the maximal numerical range is carried out. It is shown
that WeT) and Wo(TJ.'.share ..the properties -of convexity, boundedness,
compactness and connectedness; and that Wo(T) has the additional property
of closure which is not true in general for WeT). Some other properties for
which conditions must be imposed on the operator T(and sometimes on H) so that they hold true for both WeT) and Wo(T) are also considered in this
chapter.
Chapter two is concerned with the role played by the maximal numerical
range in determining the norm of a derivation on the Banach algebra, B(H).
An identity for the norm is derived using the maximal numerical range.
In chapter three, the essential maximal numerical range is discussed. This
concept is used in deriving an analogous identity for the norm of a derivation
in the quotient algebra, B(H)IK(H).
The last chapter focuses on the concepts of algebraic maximal numerical
range and the joint maximal numerical range.
It is however, necessary that before proceeding with this discussion, some
standard definitions and notation that will be adhered to herein be given.

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