On the continuity of spectra of Toeplitz operators
by
Ilya Spitkovsky
College of William and Mary
Coauthors: Albrecht Böttcher, Sergei Grudsky

Everybody knows that the mapping f : A --> \sigma(A) of operators acting on a Hilbert space to their spectra is discontinuous:

lim inf \sigma(An) subset lim sup \sigma(An) subset \sigma( lim An)

for any uniformly convergent sequence of operators, but both inclusions may be strict. However, for Toeplitz operators T(a_n),

lim inf \sigmaT(an) = \sigmaT( lim an)     (1)

whenever a uniformly convergent sequence a_n of their symbols belongs to the Douglas algebra H^\infty +C, the algebra PQC of piecewise quasicontinuous functions (D. Farenick and W. Lee, 1996), or the algebra AP+C generated by Bohr almost periodic fucntions and C (S. Hwang and W. Lee, 1998). It was conlectured by D. Farenick and W. Lee that (1) holds for any uniformly convergent sequence of L^\infty functions. We will discuss this conjecture for a_n lying in the Sarason algebra SAP of semi almost periodic functions.

Date received: April 18, 1999