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On exponential factorizations of matrices over Banach algebras. (English) Zbl 07459399

Summary: We study exponential factorization of invertible matrices over unital complex Banach algebras. In particular, we prove that every invertible matrix with entries in the algebra of holomorphic functions on a closed bordered Riemann surface can be written as a product of two exponents of matrices over this algebra. Our result extends similar results proved earlier in [7] and [8] for \(2 \times 2\) matrices.

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
15A54 Matrices over function rings in one or more variables
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[1] Brudnyi, A.; Rodman, L.; Spitkovsky, I. M., Projective free algebras of continuous functions on compact abelian groups, J. Funct. Anal., 259, 918-932 (2010) · Zbl 1204.43001
[2] Brudnyi, A., On the factorization of matrices over commutative Banach algebras, Integral Equ. Oper. Theory, 90, Article 6 pp. (2018), 8 pp. · Zbl 06859645
[3] Corach, G.; Suárez, F., Stable rank in holomorphic function algebras, Ill. J. Math., 29, 627-639 (1985) · Zbl 0606.46034
[4] Dennis, R. K.; Vaserstein, L. N., On a question of M. Newman on the number of commutators, J. Algebra, 118, 150-161 (1988) · Zbl 0649.20048
[5] Doubtsov, E.; Kutzschebauch, F., Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings, Anal. Math. Phys., 9, 1005-1018 (2019) · Zbl 1430.15010
[6] Ivarsson, B.; Kutzschebauch, F., On the number of factors in the unipotent factorization of holomorphic mappings into \(S L_2(\mathbb{C})\), Proc. Am. Math. Soc., 140, 3, 823-838 (2012) · Zbl 1250.32009
[7] Kutzschebauch, F.; Studer, L., Exponential factorizations of holomorphic maps, Bull. Lond. Math. Soc., 51, 995-1004 (2019) · Zbl 1450.30073
[8] Leiterer, J., On holomorphic matrices on bordered Riemann surfaces · Zbl 1479.30029
[9] Michael, E. A., Locally multiplicatively-convex topological algebras, Mem. Am. Math. Soc., 11 (1952) · Zbl 0047.35502
[10] Mortini, R.; Rupp, R., Reducibility of invertible tuples to the principal component in commutative Banach algebras, Ark. Mat., 54, 2, 499-524 (2016) · Zbl 1369.46040
[11] Mortini, R.; Rupp, R., Logarithms and exponentials in the matrix algebra \(\mathcal{M}_2(A)\), Comput. Methods Funct. Theory, 18, 53-87 (2018) · Zbl 1412.30145
[12] Rudin, W., Functional Analysis (1991), McGraw-Hill · Zbl 0867.46001
[13] Shilov, G. E., On the decomposition of a commutative normed ring into a direct sum of ideals, Am. Math. Soc. Transl. (2), 1, 37-48 (1955) · Zbl 0066.36103
[14] Treil, S., Stable rank of \(H^\infty\) is equal to one, J. Funct. Anal., 109, 130-154 (1992) · Zbl 0784.46037
[15] Vaserstein, L. N., Reduction of a matrix depending on parameters to a diagonal form by addition operations, Proc. Am. Math. Soc., 103, 741-746 (1988) · Zbl 0657.55005
[16] Vaserstein, L. N., Bass’s first stable range condition, J. Pure Appl. Algebra, 34, 319-330 (1984) · Zbl 0547.16017
[17] Tuni, Xandi, Logarithm of complex matrices in holomorphic families
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