Zero Sums of Idempotents and Banach Algebras Failing to be Spectrally Regular

Sums of idempotents and logarithmic residues in zero pattern matrix algebras

“…These are concerned with the situation where we have, say, the consecutive narrow transitions (12,13,14) and (20,21,22), taking for granted, of course, that n is sufficiently large. First we consider the case when they are of the same 'lower passage type':…”

“…Clearly G is irreducible. Also, the narrow passages for G are the genuine ones (0, 1, 2), (8,9,10), (9,10,11), (10,11,12), (11,12,13), (19,20,21), cf., the boldface marking in the above scheme, together with the dummies (−1, 0, 1) and (23,24,25). It is enlightening to look at the situation where m = 1, so the only narrow passages for G are the dummies (−1, 0, 1) and (n − 1, n, n + 1).…”

“…Next we focus on situation where the consecutive narrow transitions (12,13,14) and (20,21,22) are of 'lower passage type' and 'upper passage type' respectively:…”

“…Whether or not this is true in general is an open question. Papers in which sums of idempotents play a central role are [23] and [32], already mentioned above, and [2,19,[4][5][6][7][8]10,13].…”

Resolvent and logarithmic residues of a singular operator pencil in Hilbert spaces

“…where ∂D ρ is the boundary of D ρ . That is, P N ,R coincides with the right logarithmic residue of A(λ) at 0 and P N ,R with the left logarithmic residue of A(λ) at 0, see [7][8][9] and references therein for general results on logarithmic residues and idempotents.…”