On numerical ranges of generalized derivations and related properties. (English) Zbl 0537.47003

Shaw, Sen-Yen

J. Aust. Math. Soc., Ser. A 36, 134-142 (1984).

Let X, Y be two normed linear spaces and L(Y,X) denote the space of all bounded linear operators from Y to X. For \(A\in L(X)=L(X,X)\) and \(B\in L(Y)\) define \(\Delta(T)=AT-TB\) (\(T\in L(Y,X))\). Assume that S is a suitably normed invariant subspace of \(\Delta\) which contains all finite rank operators. Then the algebra numerical range of \(\Delta| S\) (the restriction of \(\Delta\) to S) is shown to be precisely the set theoretic difference of the algebraic numerical ranges of A and B. This is a numerical range analogue of L. Fialkow’s result for spectra [Trans. Am. Math. Soc. 243, 147-168 (1978; Zbl 0397.47002)].
The second result states that when \(X=Y\) and \(S=L(X)\), \(\Delta\) is Hermitian (resp. normal) if and only if A-\(\lambda\) are Hermitian (resp. normal) for some scalar \(\lambda\).
These two results have contained as special cases the results of J. Kyle [Proc. Edinb. Math. Soc. II. Ser. 21, 33-39 (1978; Zbl 0379.46061)]. An even stronger result asserts that if \(X=H\) is a Hilbert space and if S is a \(C^*\)-algebra or a minimal norm ideal in L(H), then any Hermitian (resp. normal) operator on S is of the form \(\Delta| S\) for some Hermitian (resp. normal) A and B.
The paper concludes with an application to a slight extension of S. K. Berberian’s result [Proc. Am. Math. Soc. 71, 113-114 (1978; Zbl 0388.47019)] about a Fuglede-Putnam theorem for hyponormal operators.

MSC:

47A12	Numerical range, numerical radius
47B47	Commutators, derivations, elementary operators, etc.
47B15	Hermitian and normal operators (spectral measures, functional calculus, etc.)

Keywords:

generalized derivation; Hermitian and normal operators; invariant subspace; algebra numerical range; a minimal norm ideal; Fuglede-Putnam theorem for hyponormal operators

Citations:

Zbl 0397.47002; Zbl 0379.46061; Zbl 0388.47019