On m-rectangle characteristics and isomorphisms of mixed (F)-, (DF)-spaces

In this thesis, we consider problems on the isomorphic classification and quasiequivalence properties of mixed (F)-, (DF)- power series spaces which, up to isomorphisms, consist of basis subspaces of the complete projective tensor products of power series spaces and (DF)- power series spaces. Important linear topological invariants in this consideration are the m-rectangle characteristics, which compute the number of points of the de ning sequences of the mixed (F)-, (DF)- power series spaces, that are inside the union of m rectangles. We show that the systems of m-rectangle characteristics give a complete characterization of the quasidiagonal isomorphisms between Montel spaces that are in certain classes of mixed (F)-, (DF)- power series spaces under proper de nitions of equivalence. Using compound invariants, we also show that the m-rectangle characteristics are linear topological invariants on the class of mixed (F)-, (DF)- power series spaces that consist of basis subspaces of the complete projective tensor products of a power series space of nite type and a (DF)- power series space of in nite type. From these invariances, we obtain the quasiequivalence of absolute bases in the spaces of the same class that are Montel and quasidiagonally isomorphic to their Cartesian square

On isomorphisms of spaces of analytic functions of several complex variables

In this thesis, we discuss results on isomorphisms of spaces of analytic functions of several complex variables in terms of pluripotential theoretic considerations. More specifically, we present the following result: Theorem 1 Let Ω be a Stein manifold of dimension n. Then, A(Ω) ≈ A(U[n]) if and only if Ω is pluriregular and consists of at most finite number of connected components. The problem of isomorphic classification of spaces of analytic functions is also closely related to the problem of existence and construction of bases in such spaces. The essential tools we use in our approach are Hilbert methods and the interpolation properties of spaces of analytic functions which give us estimates of dual norms and help us to obtain extendable bases for pluriregular pairs