Stevo Stevic talks with
ScienceWatch.com and answers a few questions about
this month's Fast Breaking Paper in the field of
Mathematics.

Article Title: Norm of weighted composition
operators from Bloch space to H-mu(infinity) on the unit
ball
Authors: Stevic,
S
Journal: ARS COMB
Volume: 88
Issue:
Page: :125-127
Year: JUL 2008
* Serbian Acad Sci, Math Inst, Knez Mihailova 36-3,
Belgrade 11000, Serbia.
* Serbian Acad Sci, Math Inst, Belgrade 11000,
Serbia.
(addresses have been truncated)

Why do you think
your paper is highly cited?

In recent decades, there has been a growing
interest in the study of weighted composition operators on spaces of
holomorphic functions, which are denoted by uC_{p} and
also in providing function-theoretic characterizations when the functions
u and p induce bounded or compact weighted composition
operators between spaces of holomorphic functions.

Usually, some necessary and sufficient
conditions for the operator to be bounded or compact are given, sometimes
with an asymptotic formula for its operator norm. In this short note, we
managed to calculate the norm of the weighted composition operator from the
Bloch space to a weighted-type space on the unit ball. The problem of
calculating operator norms is a basic one but usually quite difficult.
There are not many papers which contain such results. This is one of the
reasons why my paper is of particular interest.

Does it describe a
new discovery, methodology, or synthesis of knowledge?

Working on Bloch-type spaces, I've realized
that sometimes it is useful to use the norm on the space by taking radial
derivatives, while in some other cases it's better to use the norm
containing the gradient of the functions in the space—as in the
current paper.

In calculating operator norms of weighted
composition operators, it turned out that it is important to obtain an
exact estimate for the point evaluation operator, for getting an estimate
of the operator norm from above and in choosing some test functions which
have growth approximately equal to the estimate, to obtain an estimate of
the operator norm from below. This need not lead to one's getting a formula
for operator norm of a weighted composition operator, but it does offer a
method which can be considered useful.

Would you summarize
the significance of your paper in layman's terms?

This paper reestablished interest in an area
not only connected to weighted composition operators, but also to other
operators on spaces of holomorphic functions. The method seems fruitful
indeed. For example, in my recent paper: "Norms of some operators from
Bergman spaces to weighted and Bloch-type space," Util. Math. 76:
59-64, 2008, among others, I calculated the norm of some integral-type
operators, while in "Norm and essential norm of composition followed by
differentiation from
a-Bloch
spaces to
,"
Appl. Math. Comput. 207: 225-29, 2009, I calculated the norm
of composition followed by differentiation operator from the Bloch and
the little Bloch space to a weighted-type spaceon the
unit disk and gave an upper and a lower bound for the essential norm
of the operator from the
a-Bloch
space to.

How did you become
involved in this research, and were there any problems along the
way?

The main result in my paper was motivated by
the results shown in my article entitled: "Weighted composition operators
between H(infinity) and
a
-Bloch spaces in the unit ball" Taiwanese J. Math. 12:
1625-39, 2008, and is a kind of addendum to it. I had initially hesitated
to publish the note because of its brevity, but in a private communication
with several experts, I realized that it could actually be quite
interesting.

In general, I first became involved in this
theory about 10 years ago by reading papers on the research area as
published in several journals. It is well-known that I'm a self-taught
mathematician and, having been on my own, I was simply looking for areas
which could be of special interest to me. I then came across several
fascinating articles on the subject of composition operators, and this led
to my examination of the field.

Where do you see
your research leading in the future?

There are many interesting areas in
mathematics and it is well known that I've published papers across several
quite different fields, so it's difficult to predict which direction may
prevail in my future research. I will certainly continue to study
systematically the properties of various operators on the spaces of
holomorphic functions.

Recently, I introduced several new
integral-type operators which have already attracted some attention and I
will continue to study recent types of nonlinear difference equations,
which are not closely connected to differential equations, such as,
rational and max-type difference equations. I am also interested in
equations which model some real-life situations.

Stevo Stevic
Mathematical Institute of the Serbian Academy of Sciences
Belgrade, Serbia