Estimates of operator polynomials in the space Lp with respect to the multiplier norm

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. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 133–148, 1976. The author wishes to express his thanks to N. K. Nikol'skii for suggesting the problem and for his interest in the paper.