On polynomial connections between projections
type: Item
instance of: scholarly article
title: On polynomial connections between projections
zbMATH Open document ID: 0714.47011
DOI: 10.1016/0024-3795(89)90007-4
author: Jaroslav Zemánek
author: E. jun. Makai
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: A result of \textit{M. Tremon} [Linear Algebra Appl. 64, 115-132 (1985; Zbl 0617.46054)] is used to show that any two Banach space projections of the same finite rank can be connected by a projection-valued polynomial path of degree not exceeding 3. It is not known whether this result can be extended to Banach algebras. An affirmative answer would follow from a spectral conjecture arising in \textit{B. Aupetit}, \textit{T. J. Laffey} and \textit{J. Zemanek} [Linear Algebra Appl. 41, 131-135 (1981; Zbl 0471.46030)]. But the authors construct two similar projections P and Q, acting on a separable Hilbert space, such that 1 is an eigenvalue of \(A+B\) for all operators A and B with \(\| P-A\| <1\) and \(\| Q- A\| <1\), which disproves this conjecture.
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 4175784
zbMATH Keywords: similarity
zbMATH Keywords: projection-valued polynomial path
zbMATH Keywords: similar projections
reviewed by: Marinus Kaashoek
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90007-4
OpenAlex ID: W2092158992
cites work: Spectral classification of projections
cites work: Q3699449
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cites work: Perturbation theory for nullity, deficiency and other quantities of linear operators
cites work: Q4173837
cites work: An elementary proof of a fundamental theorem in the theory of Banach algebras
cites work: Polynômes de degré minimum connectant deux projections dans une algèbre de Banach. (Polynomials of minimal degree connecting two projections in a Banach algebra)
cites work: Idempotents in Banach Algebras
description: scientific article; zbMATH DE number 4175784