For and an element of a complex unital Banach algebra , we prove the following two topological properties about the level sets of the condition spectrum. (1) If , then the -level set of the condition spectrum of has an empty interior unless is a scalar multiple of the unity. (2) If , then the -level set of the condition spectrum of has an empty interior in the unbounded component of the resolvent set of . Further, we show that, if the Banach space is complex uniformly convex or if is complex uniformly convex, then, for any operator acting on , the level set of the -condition spectrum of has an empty interior.
"Level sets of the condition spectrum." Ann. Funct. Anal. 8 (3) 314 - 328, August 2017. https://doi.org/10.1215/20088752-0000016X