On the Numerical Radius and Its Applications*

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finiteand infinitedimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems. 1. DEFINITION, BOUNDS, AND EVALUATION Let H be a Hilbert space over the complex field C with inner product (x,y) and norm llxll=(x,x) ‘/’ Let C&(H) denote the algebra of bounded linear . operators on H. Then, for any A in %3(H) we define the numerical radius of *Notes solicited by Professor Morris Newman, following a series of lectures delivered by the first author at the Institute for the Interdisciplinary Applications of Algebra and Combinatorics, University of California, Santa Barbara, California, Summer 1980. ‘Research sponsored in part by the Air Force Office of Scientific Research, Air Force System Command, United States Air Force Grant Nos. AFSOR-763046 and AFSOR-799127. LINEAR ALGEBRA ANDITSAPPLZCATZONS42:263-284 (1982) 263 0 Elsevier Science Publishing Co., Inc., 1982 52 Vanderbilt Ave., New York, NY 10017 0024-3795/82/010263 + 22$02.75 264 A. MOSHE GOLDBERG AND EITAN TADMOR r(A)=sup{~(Ar,r)~:~~~~~=l}. Three of the most obvious properties of r are: r(A)>@ r(oA)=]a]r(A) V’aEC; r(A+B)+A)+r(B) VA, BE%(H). (1.1) (1.2) (1.3) In this section we discuss further bounds for r as well as ways to evaluate it. THEOREM 1.1 (e.g. [ll], [18]). Let p(A) be the spectral radiu.s ofA. Then