A Simultaneous Lifting Theorem for Block Diagonal Operators

Stampfli has shown that for a given T £ B(H) there exists a K £ C(H) so that o(T + K) = ow(T). An analogous result holds for the essential numerical range We(T). A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator T £ B(H) if o(T + K) = o„(T) and W(T + K)= We(T). Theorem. For each block diagonal operator T, there exists a compact operator K which preserves the Weyl spectrum and essential numerical range of T. The perturbed operator T + K is not, in general, block diagonal. An example is given of a block diagonal operator T for which there can be no block diagonal perturbation which preserves the Weyl spectrum and essential numerical range of T. Let 77(77) and C(77) denote, respectively, the algebras of bounded and compact linear operators on a complex, separable Hubert space 77. Then C(H) is a closed ideal in B(H) and 77(77)/ C(77), the Calkin algebra, is a C*-algebra with identity when endowed with the quotient norm. One general problem associated with this algebra is the following: if a coset T + C(H) has a certain property in 77(77)/C(77), is there a representative T + K of the coset having the same property in 77(77)? Much progress on this question has already been made (see, for instance, [1], [2], [5], [6], [9], [10], [11]). In particular, Stampfli [11] has shown that there exists C G C(77) such that the spectrum of T + C and the Weyl spectrum of T are equal. In [6], it was proved that there is a C G C(H) such that the closure of the numerical range of T + C agrees with the essential numerical range of T. The results in the present paper were motivated by the following question: Given T E 77(77), does there exist a C G C(77) such that T + C simultaneously preserves the Weyl spectrum and essential numerical range of T. This problem appears to be quite hard and is still unresolved. Our main result is Theorem 3.7. For each block diagonal operator T G 77(77), there exists a compact operator C such that T + C simultaneously preserves the Weyl spectrum and essential numerical range of T. Received by the editors May 7, 1978 and, in revised form, July 6, 1978. AMS (MOS) subject classifications (1970). Primary 47A10; Secondary 47B05.