Application of entropy theory to the problem of data compression

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(1). New locally convex topologies constructed with some families of nuclear operators on a separable Hilbert space which are weaker than the norm topology and stronger than the weak topology are given. These topologies are applied to the classification of nuclear operators. Further, the homeomorphism problem of subspaces with the norms in terms of the ranges of the closed unit ball under nuclear operators is treated. This result is applied to the homeomorphism problem of periodic and continuously differentiable function spaces included by LィイD12ィエD1[0,1].(2). The homeomorphism problems of subspaces with norms in terms of the ranges of the closed unit ball under compact positive operators are examined. These results will be applied to the operator theoretical classification of reproducing kernel Hilbert spaces.(3). It is shown that Ohya's entropy dimension on a WィイD1*ィエD1-dynamical system is time-invariant, and homeomorphism problems of operator algebras equipped with the quasi σ-strong operator topologies are discussed.(4). A formula for estimating ε-entropy of a compact positive operator in terms of the distribution of proper values of such an operator was given by Prosser and Root. In this paper, an inversion formula for estimating the distribution of proper values of a compact positive operator in terms of ε-entropy of such an operator is given.(5). The homeomorphism problem of compact nonlinear mappings on a locally convex topological vector space is studied under the method of dimension theory. First of all, dimension theoretic homeomorphism invariants which are defined on the set of all compact nonlinear mappings are introduced. Next, these invariants are applied to dimension theoretic characterization of the fixed point sets which these compact nonlinear mapping have.