The modulus of a regular linear operator and the “zero-two” law in Lp-spaces (1 < p < + ∞, p ≠ 2)

On the “zero-two” law for positive contractions

“…The next proposition is similar to Proposition 4.2 of [4]. As expected, in the case of an 1} -space the statement is stronger.…”

“…If we assume that Vy> is a positive contraction, then in the case of an I^-space (Proposition 2.1) we obtain that ||K < / ) ||^2 for every deN, while in the case of an IS -space, 1 < p < + oo given in Proposition 3.1 of [4], the upper bound for the sequence HIIXfeN depends on £ (tends to + oo as t tends to + oo).…”

“…As expected, in the case of an 1} -space the statement is stronger. Proof It is known (see the proof of Theorem B in Section 4 of [4]) that using Stirling's formula, one can find a positive constant }>>0 such that for every Banach lattice E, for every positive contraction T of £ and for every £ e N Now let E be an L 1 -space, let T be a positive contraction of E and let m e N u {0} be such that ||7"»+i_(7"»+i A T m )||<l.…”

The strong ``zero-two" law for positive contractions of Banach--Kantorovich $L_p$-lattices

“…An extension of this result to positive operators on L ∞ -spaces was given by Foguel [3]. In [27] Zahoropol generalized these results, calling it the "zero-two" law, and his result can be formulated as follows:…”

ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES

“…In this paper we continue the previous investigations and prove a multiparametric generalization of the uniform "zero-two" law in L 1 -space. Note that a different kind of generalization of the said law is given in [8,15,19].…”