The following notation is used throughout the paper: N is the set of all natural numbers. R is the set of all real numbers, R+ = [0,+∞[,[x]+ = (1/2)(|x|+ x), [x]− = (1/2)(|x|− x). C([a,b];R) is the Banach space of continuous functions u : [a,b]→ R with the norm ‖u‖C =max{|u(t)| : t ∈ [a,b]}. C̃([a,b];R) is the set of absolutely continuous functions u : [a,b]→ R. L([a,b];R) is the Banach space of...

CONTINUOUS SPECTRA. II BY C. R. PUTNAM Communicated by Maurice Heins, November 1, 1967 1. This paper is a continuation of [4]. It may be recalled that if A is a self-adjoint operator on a Hubert space § with spectral resolution A=zf\dE\, then the set of elements x in § for which ||-Ex#|| is an absolutely continuous function of X is a subspace, &a(A), of § (see, e.g., Halmos [l, p. 104]). The op...

Let A be a certain irreducible 0-1 matrix and let t denote the family of piecewise linear Markov maps on [0,1] which are consistent with A. The main result of this paper characterizes those maps in t whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with A, the absolutely continuous invariant measure is not maximal.

In this paper, we study the stability of the absolutely continuous spectrum of onedimensional Stark operators under various classes of perturbations. Stark Schrödinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather ...

Throughout the paper L1[a,b] denotes the set of integrable functions on [a,b], Lloc(a,b] the set of functions x : (a,b]→ R which are integrable on [a− ε,b] for arbitrary small ε > 0, AC[a,b] is the set of absolutely continuous function on [a,b], and ACloc(a,b] is the set of functions x : (a,b]→ R which are absolutely continuous on [a− ε,b] for arbitrary small ε > 0. Let T be a positive number. ...

For certain families of compact subsets the plane, isomorphism class algebra absolutely continuous functions on a set is completely determined by homeomorphism set. This analogous to Gelfand–Kolmogorov theorem for C(K) spaces. In this paper, we define family sets comprising finite unions convex curves and show that has ‘Gelfand–Kolmogorov’ property.