Block diagonalization in Banach algebras

Author:
Robin Harte

Journal:
Proc. Amer. Math. Soc. **129** (2001), 181-190

MSC (1991):
Primary 47A13; Secondary 15A21, 15A18

DOI:
https://doi.org/10.1090/S0002-9939-00-05884-6

Published electronically:
August 17, 2000

MathSciNet review:
1784022

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

``Reduction" of linear operators is effected by commuting projections; the spectrum of the operator is then the union of the spectra of its range and null space restrictions. Disjointness of these partial spectra implies that the projection ``double commutes" with the operator, which in turn can be recognised as a curious kind of ``exactness". Variants of this exactness correspond to various kinds of disjointness between the partial spectra.

**1.**Hong-Ke Du and Jin Pan,*Perturbation of spectrums of operator matrices*, Proc. Amer. Math. Soc.**121**(1994) 761-766. MR**94i:47004****2.**Jin Kyu Han, Hong Youl Lee and Woo Young Lee,*Invertible completions of upper triangular operator matrices*, Proc. Amer. Math. Soc.**128**(2000) 119-123. MR**2000c:47003****3.**R.E. Harte,*Invertibility and singularity*, Dekker New York 1988. MR**89d:47001****4.**R.E. Harte,*Invertibility and singularity for operator matrices*, Proc. Royal Irish Acad.**88A**(1988) 103-118. MR**90a:15019****5.**R.E. Harte,*Unspectral sets*, Rendiconti del Circ. Mat. Palermo**56**(1998) 69-77. CMP**2000:01****6.**R.E. Harte and C. Hernandez,*On the Taylor spectrum of left-right multipliers*, Proc. Amer. Math. Soc.**126**(1998) 397-404. MR**98e:46057****7.**K. Hoffman and R. Kunze,*Linear algebra*, Prentice Hall New York 1971. MR**43:1998****8.**J.J. Koliha,*Block diagonalization*, Mathematica Bohemica (to appear).**9.**R.R. London and H.P. Rogosinski,*Decomposition theory in the teaching of elementary linear algebra*, Amer. Math. Monthly**97**(1990) 478-485. MR**91d:15022****10.**H. Radjavi and P. Rosenthal,*Invariant subspaces*, Springer New York 1973. MR**51:3924****11.**Ch. Schmoeger,*Remarks on commuting exponentials on Banach algebras*, Proc. Amer. Math. Soc.**127**(1999) 1337-1338. MR**99h:46090****12.**D. A. Suprunenko and R.I. Tyshkevich,*Commutative matrices*, Academic Press New York 1968.**13.**J. H. M. Wedderburn,*Lectures on matrices*, AMS Colloq. Publ.**17**, Amer. Math. Soc. Providence 1934.**14.**E. M. E. Wermuth,*A remark on commuting operator exponentials*, Proc. Amer. Math. Soc.**125**(1997) 1685-1688. MR**97g:39011**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47A13,
15A21,
15A18

Retrieve articles in all journals with MSC (1991): 47A13, 15A21, 15A18

Additional Information

**Robin Harte**

Affiliation:
School of Mathematics, Trinity College, Dublin 2, Ireland

Email:
rharte@maths.tcd.ie

DOI:
https://doi.org/10.1090/S0002-9939-00-05884-6

Keywords:
Commuting idempotent,
double commutant,
spectral disjointness,
exactness conditions

Received by editor(s):
December 15, 1997

Received by editor(s) in revised form:
March 10, 1998, October 6, 1998, and March 31, 1999

Published electronically:
August 17, 2000

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society