Approximately Finite Dimensional with Differentiability of Lipschitz Functions on Banach Spaces and Point Spectra of Power Bounded Operators.

Abstract

We show some Banach spaces of measurable operator-valued functions and established an interpolation formula for the measure of noncompactness. We study the spectral theory of ordered pairs of the linear operators on the Lebesgue space on the real line and continuous spaces on a compact Hausdorff space and indifferent Banach spaces . We investigate the cyclicity and unicellular- rity of the differentiation and nonwandering operators on Banach spaces of formal power series. We give the Frechet differentiability of convex Lipschitzian functi- ons on a Banach space with Gaussian measures . We show the construction of the approximately finite-dimensional Banach spaces and study spectral theory with complex interpolation to determine size estimates. We characterize the closed commutants and equivalence of generalized norms of the Backward shift operators. We show the duality of Hardy space and differentiability with a prevalent transver- sality result of Lipschitz functions on normed and Banach spaces. We study examples of unbounded and bounded imaginary powers of certain differential operators with conical singularities . Especial point spectra of partially power-bounded operators are considered .