Heinz’s inequality and Bernstein’s inequality

Abstract

The purpose of the present account is to sharpen Heinz's inequality, and to investigate the equality and the bound of the inequality. As a consequence of this we present a Bernstein type inequality for nonselfadjoint operators. The Heinz inequality can be naturally extended to a more general case, and from which we obtain in particular Bessel's equality and inequality. Finally, Bernstein's inequality is extended to n eigenvectors, and shows that the bound of the inequality is preserved. The well-known Heinz inequality is as follows: The relation H8 I(Tx,y) 12 < (IT 12ax x)(IT* 12(1-,)y, y) holds for any bounded linear operator T on a complex Hilbert space H, x, y E H, and any real number oa with 0 < a < 1, where IT] is the positive square root of the operator T*T. It is possible to sharpen the inequality (*) if T*y is orthogonal to a vector z with Tz $4 0. The new inequality is naturally extended to a more general case when T*y is orthogonal to a set of vectors in which the bound of the inequality is retained. In particular we obtain Bessel's equality. By a similar method we present a Bernstein type inequality for nonselfadjoint operators. Finally, Bernstein's inequality is generalized to n eigenvectors for a selfadjoint operator and shows that the bound is preserved. Theorem 1. Let T be a bounded linear operator on a complex Hilbert space H and 0 5$ y E H. If T*y is orthogonal to a vector z E H with Tz $ 0, then I(TX I y)12 + 2( T*2(I -a)y, y)(IT] 2a X, Z) 1j2 < (IT I2X, x) (IT* 12(1-,a) If XY)I + ~~(IT12cZ, Z)-( Ixx y for every x E H and a E [0,1]. The equality holds if and only if the two vectors T*y and IT 12caX_(IT )I(ITI12 a z) are proportional, equivalently, the two vectors Tx (jTjl2xz)Tz and IT*12(1)y are proportional for 0 < a < 1. (ITI 2 Z, z) ITI()y pootnafr Received by the editors September 7, 1995 and, in revised form, September 19, 1995 and February 5, 1996. 1991 Mathematics Subject Classification. Primary 47A30, 65F15; Secondary 65J10. ?)1997 Arnerican Mathematical Society 2319 This content downloaded from 207.46.13.113 on Thu, 06 Oct 2016 04:34:08 UTC All use subject to http://about.jstor.org/terms