The uniform zero-two law for positive operators in Banach lattices

• Autorzy: Michael Lin
• Języki publikacji: EN

Abstrakty

EN

Let T be a positive power-bounded operator on a Banach lattice. We prove: (i) If $inf_n ||T^n(I-T)|| < 2$, then there is a k ≥ 1 such that $lim_{n→∞} ||T^n(I-T^k)|| = 0. (ii) $lim_{n→∞} ||T^n(I-T)|| = 0$ if (and only if) $inf_n ||T^n(I-T)|| < √3$.

Kategorie tematyczne

• 47A35: Ergodic theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 131
• Numer: 2
• Strony: 149-153

Daty

wydano

1998

otrzymano

1997-08-27

Twórcy

autor

Michael Lin

• Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Bibliografia

• [AR] G. R. Allan and T. J. Ransford, Power-dominated elements in Banach algebras, Studia Math. 94 (1989), 63-79.
• [B] D. Berend, A note on the $L^p$ analogue of the "zero-two" law, Proc. Amer. Math. Soc. 114 (1992), 95-97.
• [F] S. R. Foguel, More on the "zero-two" law, ibid. 61 (1976), 262-264.
• [FWe] S. R. Foguel and B. Weiss, On convex power series of a conservative Markov operator, ibid. 38 (1973), 325-330.
• [KT] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 312-328.
• [M] J. Martinez, A representation lattice isomorphism for the peripheral spectrum, Proc. Amer. Math. Soc. 119 (1993), 489-492.
• [OSu] D. Ornstein and L. Sucheston, An operator theorem on $L^1$ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631-1639.
• [S1] H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974.
• [S2] H. Schaefer, The zero-two law for positive contractions is valid in all Banach lattices, Israel J. Math. 59 (1987), 241-244.
• [Sc] A. Schep, A remark on the uniform zero-two law for positive contractions, Arch. Math. (Basel) 53 (1989), 493-496.
• [W1] R. Wittmann, Analogues of the zero-two law for positive contractions in $L^p$ and C(X), Israel J. Math. 59 (1987), 8-28.
• [W2]] R. Wittmann, Ein starkes ``Null-Zwei"-Gesetz in $L^p$, Math. Z. 197 (1988), 223-229.
• [Z1] R. Zaharopol, The modulus of a regular operator and the "zero-two" law in $L^p$-spaces (1 < p < ∞, p ≠ 2), J. Funct. Anal. 68 (1986), 300-312.
• [Z2] R. Zaharopol, Uniform monotonicity of norms and the strong "zero-two" law, J. Math. Anal. Appl. 139 (1989), 217-225.
• [Ze] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 369-385.