APPROXIMATION PROPERTIES AND SOME CLASSES OF
OPERATORS

I. Approximation properties, tensor products and operator ideals

Here we present the project of investigations of the relations between the theory of. the operator ideals (in sense of A. Pietsch), the theory of tensor products of Defant and Floret and different approximation properties introduced by A. Grothendieck, P. Saphar, O. Reinov and others.

The theories of normed operator ideals and normed tensor product play very important role in many fields of Functional Analysis; they have their great applications, e.g., in spectral theory of operators, in the geometry of both finite dimensional and infinite dimensional Banach spaces, in the theory of probabilities One of the main connection between these two theories is given by different approximation conditions (in Banach spaces), considered by many authors from A.Grothendieck, P.Saphar, Oleg Reinov till Defant, Floret, Cassaza and many other mathematicians.

Since P.Enflo has constructed his famous counterexample to a problem of A.Grothendeck
(and to the basis problem of S.Banach) the interest to the theme constantly (till nowadays)
increases. Still now there are a lot of open problems in the mentioned areas of Functional
Analysis.

II. Applications of the theories

Many applications of the theories concern such problems as the problems of good factorizations of operators from different classes (ideals), problems of isomorphic classifications of Banach spaces (e.g. with the help of the theory of operator ideals it was possible to prove that the classical Disk-algebra A and the space C(T) are not isomorphic). A lot of applications were given in the questions of the distributions of the eigenvalues of operators. In many such questions it was important to know whether or not Banach spaces under consideration possesses some approximation properties. The last questions are closely connected with the questions of type “is it true that if a dual operator is “good” that the operator itself is also “good”?’ Many contributions in solving such problems were brought by O.Reinov. For example, he has shown that it does not follow from p-nuclearity of the second dual operator the pnuclearity of the operator itself - even in the spaces with bases (l<p<2)! See [1], [2]

III. Perspectives

We plan to develop the techniques of [1],[2] (and others) using one of the modification of a famous theorem of J.Lindenstrauss to close the last problem for cases p>2. We plan also obtain (by the same way) some generalizations of the result to the cases of lp-factorable operators.

The investigations of the project can be done in the frame of cooperation of these universities and research institutes:
School of Mathematical Sciences (Lahore, Pakistan);
S-Petersburg State University (S-Petersburg, Russia);
S-Petersburg Branch of the Mathematical Institute RAN (S-Petersburg, Russia).

References:
1. Reinov O.I. Approximation properties and some classes of operators // Problemy matematicheskogo analiza, Novosibirsk, 2001, vol. 23, 147-205.
2. Reinov O.I.. Excursus to the approximation theory of operators in operator ideals // Problems of nowadays approximation theory, St. Petersburg, SPb GU, 2004, 231-293.