Non-commutative standard polynomials applied to matrices

Abstract : The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra M n (K). For K = R or C and 2 ≤ r ≤ 2n − 1, we investigate how big S r (A 1 ,. .. , A r) can be when A 1 ,. .. , A r belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball when r = 2 or 3 and n = 2. MSC classification : 15A24, 15A27, 15A60
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Submitted on : Thursday, November 24, 2016 - 3:46:21 PM
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Denis Serre. Non-commutative standard polynomials applied to matrices. Linear Algebra and its Applications, Elsevier, 2016, 490, pp.202 - 223. ⟨10.1016/j.laa.2015.11.003⟩. ⟨ensl-01402364⟩

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