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On polynomial connections between projections. (English) Zbl 0714.47011

A result of 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 B. Aupetit, T. J. Laffey and 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.
Reviewer: M.A.Kaashoek

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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References:

[1] Aupetit, B.; Laffey, T. J.; Zemánek, J., Spectral classification of projections, Linear Algebra Appl., 41, 131-135 (1981) · Zbl 0471.46030
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