The Browder Spectrum of an Elementary Operator

Abstract

We relate the ascent and descent of n-tuples of multiplication operators $M_{a,b}(u) = aub$ to that of the coefficient Hilbert space operators a, b. For example, if $a = (a_1,\ldots,a_n)$ and $b^* = (b^*_1,\ldots,b^*_m)$ have finite non-zero ascent and descent s and t, respectively, then the $(n + m)$-tuple $(L_a, R_b)$ of left and right multiplication operators has finite ascent and descent $s + t - 1.$. Using these results we obtain a description of the Browder joint spectrum of $(L_a, R_b)$ and provide formulae for the Browder spectrum of an elementary operator acting on B(H) or on a norm ideal of B(H).

Mathematics Subject Classification (2000).Primary 47B49; Secondary 47A13