Estimates of operator polynomials in the space L^p with respect to the multiplier norm

Abstract

The paper is devoted to the determination of an analog of J, von Neumann's inequality for the space L^p.The fundamental result of the paper is: If T is an absolute contraction in the space L^p(X,F,μ) (i.e., |T|_L1≤1) and |T|_L∞≤1) then for every polynomial ϕ one has

where S is the shift operator in the space. On the basis of this theorem, one finds a theorem on substitutions in the space of multipliers. One gives applications of the inequality (1) to the weighted shift operators in the space It turns out that under some natural restrictions on the weight, inequality (1) becomes an equality for such operators. One also presents a proof of J. von Neumann's inequality based on the approximation of a contraction in a Hilbert space by unitary operators.