On Symmetric Operator Ideals and s-Numbers
Motivated by the well-known theorem of Schauder, we study the relationship between various s-numbers of an operator T and its adjoint T∗ between Banach spaces. For non-compact operator T ∈ L(X, Y ), we do not have a lot of information about the relationship between n-th s-number, sn(T), with sn(T∗ ), however, in chapter 2, by considering X and Y , with lifting and extension properties, respectively, we were able to obtain a relationship between sn(T) with sn(T∗ ) for certain s-numbers. Using a certain characterization of compactness together with the Principle of Local Reflexivity, we give a different simpler proof of Hutton’s theorem. In chapter 3, by considering operators which are not compact but compact with respect to certain approximation schemes Q, which we call Q-compact, we proved Hutton’s Theorem for Q-compact operator T and symmetrized approximation numbers, which answers the question of comparing the degree of compactness for T and its adjoint T∗ for noncompact T. Chapter 4 defines the K-functional via rearrangement-invariant function spaces, studies its effect on interpolation spaces, applies interpolation theory to some linear and non-linear partial differential equations, and also gives some criteria for the boundedness of the norms of operators arising from PDEs in some concrete Banach spaces. Under natural conditions regarding Bernstein and Jackson inequalities, interpolation spaces can be realized as approximation spaces. Consequently, the final chapter 5 defines approximation spaces for compact H-operators using the sequences of their eigenvalues and establishes relations among these spaces using interpolation theory, and presents an inclusion theorem and a representation theorem.
Approximation schemes, Approximation spaces, Compact operators, Interpolation spaces, s-numbers, Schauder's theorem, Mathematics
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