On Shift Operators

Abstract

A definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.

Some remarks on distributional chaos for bounded linear operators

“…Without loss of generality we may assume f (0) = 0 ; otherwise, take h : [12]. Since f is continuous at 0 , for any τ > 0 there exists δ > 0 such that for any x ∈ X , ∥x∥ < δ implies ∥f (x)∥ < τ .…”

Isometric Shifts and Metric Spaces

“…Our example shows somehow that D can be far from being dense in X in the sense that X \ cl X (D) is uncountable. Our X also has, contrary to what Holub conjectured in [9], an infinite connected component (see also [6,Corollary 2.1…”

“…In [6], Gutek, Hart, Jamison and Rajagopalan extended many of the results obtained by J.R. Holub in [9] concerning isometric shift operators on the Banach space C(X) (X compact Hausdorff). First, they classified codimension 1 linear isometries on C(X) using the following result: let T : C(X) −→ C(X) be a codimension 1 linear isometry.…”

A note on finite codimensional linear isometries of C(X) into C(Y)

“…In ], A. Gutek, D. Hart, J. Jamison and M. Rajagopalan proved that there are no isometric shift operators on C([a,b]), a result first proved in the real scalars by Holub [3]. Here [a,b] is any closed interval in the real line and C( [a, b]) is the Banach space of all continuous complex-valued functions on [a,b].…”