Numerical Radius Perserving Operators on B(H)

Abstract

Let H be a Hilbert space over @ and let B(H) denote the vector space of all bounded linear operators on H . We prove that a linear isomorphism T : B(H) + B(H) is numerical radius-preserving if and only if it is a multiply of a C*-isomorphism by a scalar of modulus one. Let H be a Hilbert space over @ and let B(H) denote the vector space of all bounded linear operators on H . For every A in B(H) , the numerical range and the numerical radius of T are defined respectively by w ( A ) = { ( ~ x , x ) : x ~ H , Ilxll = 1) , w(A) = sup { l A :2 E W(A)) . It is well known that w(.) is a norm on B(H) and that this norm is equivalent to the usual operator norm. (See [4, p. 1171.) A classical theorem of Kadison [4, Theorem 71 asserts that every linear isomorphism on B(H) which is isometric with respect to the operator norm is a C*-isomorphism followed by left multiplication by a fixed unitary operator. A C*-isomorphism is a linear isomorphism of B(H) such that T(A*)= T(A)* for all A in B(H) and T(An)= for all selfadjoint A in B(H) and all natural number n . A description of C*-isomorphisms on B(H) can be obtained. First of all we have from [6, Corollary 111 that a C*-isomorphism on B(H) is either a *isomorphism or a *-anti-isomorphism. Suppose that T is an algebra isomorphism on B(H) . Then by [3, Theorem 21, there is an invertible operator V on H such that T(A) = VA V-' for all A in B(H) . If we also assume that T(A*)= T(A)* for all A in B(H) , then VA* V-' = (V-')*A* V* and hence (V* V)A* = A*(V* V) for all A in B(H) . It follows that V* V is a scalar multiple of the identity operator I . Say V* V = k l . As V* V is always a positive operator and k cannot be zero, k > 0 . Let U = L V . Then U is unitary Ji; and T(A) = UAU* for all A in B(H) . For a *-anti-isomorphism T , it can be shown (e.g., see [5, Remark 21) that there is a unitary operator U in B(H) such that T(A) = UA'U* for all A in B(H) ,where A' denotes the transpose Received by the editors June 15, 1993 and, in revised form, August 2, 1993. 1991 Mathematics Subject Classification. Primary 47B49, 47A12. @ 1995 American Mathematical Society 0002-9939195 $1.00 + S.25 per page