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Ergodic Theorems for Group Actions pp 10–50Cite as

Means and Averageable Functions

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 78))

Abstract

In this chapter we use the following notations: X is a semigroup with identity e; B is a Banach space; ¦µ B is the Banach space of all bounded functions f: X 1 ↦ B with the norm ||f ||¦µB:=supx¡ÊX ||f£¨x£©||B, f ¡Ê ¦µB; \(\tilde P$% MathType!End!2!1!\) is the set of all discrete probability measures with finite carriers (it is well known that \(\tilde P$% MathType!End!2!1!\) is a_semigroup with respect to convolution). We associate with any ¦Ë ¡Ê \(\tilde P$% MathType!End!2!1!\) two averaging operators “with weight ¦Ë ”: the right one, R¦Ë, and the left one, L¦Ë:

$$\left( {{R_\lambda }f} \right)\left( x \right) = \smallint f\left( {xy} \right)\lambda \left( {dy} \right) = \sum\limits_{i = 1}^k {{\alpha _i}} f\left( {x{x_i}} \right)$$
(1)
$$\left( {f{L_\lambda }} \right)\left( x \right) = \smallint f\left( {yx} \right)\lambda \left( {dy} \right) = \sum\limits_{i = 1}^k {{\alpha _i}} f\left( {{x_i}x} \right)$$
(2)

where {x1,¡­,xk} = c(¦Ë), the carrier of ¦Ë, and ¦Ái = ¦Ë({xi}). In particular, if ¦Ë = ¦Äxo is the measure concentrated at the point x0, then the operators \({R_{xo}}: = {R_{{\delta _{xo}}}}\) and \({L_{xo}}: = {L_{{\delta _{xo}}}}\) are the right and the left translation in ¦µ B , respectively, that is, (R xo )(x) = f(xx o ) and (f L x0 )(x) = f (x 0 x). It is very easy to verify the following properties:

  1. (i)

    R ¦Ë (f) = L ¦Ë (f)= f if f(x)=b,b¡ÊB;

  2. (ii)

    R ¦Ë and L ¦Ë are linear contractions in ¦µB£¬¦Ë¡Ê \(\tilde P$% MathType!End!2!1!\);

  3. (iii)

    R ¦Ë1 R ¦Ë2 = R¦Ë1 *¦Ë2, L ¦Ë1 L ¦Ë2 = L ¦Ë1 * ¦Ë2 , that is, both R: ¦Ë ¡ú R¦Ë and L: ¦Ë¡ú L ¦Ë are representations of \(\tilde P$% MathType!End!2!1!\) in ¦µ B , R ¦Ë being a left one and L¦Ë being a right one;

  4. (iv)

    the operators R¦Ë1, and L ¦Ë2 commute, i.e. R ¦Ë 1,L ¥»2 = L¦Ë2,R¦Ë1 ¦Ë1 ¦Ë2 ¡Ê \(\tilde P$% MathType!End!2!1!\).

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  1. Department of Statistics, Penn State University, University Park, USA

    Arkady Tempelman

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© 1992 Springer Science+Business Media Dordrecht

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Tempelman, A. (1992). Means and Averageable Functions. In: Ergodic Theorems for Group Actions. Mathematics and Its Applications, vol 78. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1460-0_2

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  • DOI: https://doi.org/10.1007/978-94-017-1460-0_2

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