Abstract
LetV be ann-dimensional inner product space,T i ,i=1,...,k, k linear operators onV, H a subgroup ofS m (the symmetric group of degreem), χ a character of degree 1 andT a linear operator onV. Denote byK(T) the induced operator ofT onV χ(H), the symmetry class of tensors associated withH and χ. This note is concerned with the structure of the setK Hχ, m (T1,...,Tk) consisting of all numbers of the form traceK(T 1 U 1...T k U k ) whereU i ,i=1,...k vary over the group of all unitary operators onV. For V=ℂn or ℝn, it turns out thatK Hχ, m (T1,...,Tk) is convex whenm is not a multiple ofn. Form=n, there are examples which show that the convexity of Hχ, m (T1,...,Tk) depends onH and χ.
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References
Horn, A.: On the singular values of a product of completely continuous operators. Proc. Nat. Acad. Sci. USA36, 374–375 (1950).
Horn, A.: On the eigenvalues of a matrix with prescribed singular values. Proc. Amer. Math. Soc.5, 4–7 (1954).
Marcus, M.: Finite-Dimensional Multilinear Algebra, Part I. New York: Marcel Dekker. 1973.
Tam, T. Y.: A unified extension of some results of Thompson, Marcus and Moyls. Mh. Math.98, 157–165 (1984).
Thompson, R. C.: The bilinear field of values. Mh. Math.81, 153–167 (1976).
Thompson, R. C.: Singular values, diagonal elements and convexity. SIAM J. Appl. Math.32, 39–63 (1977).
Weyl, H.: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. USA35, 408–411 (1949).
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The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for his valuable advice and encouragement.
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Tam, TY. Induced operators on symmetry classes of tensors. Monatshefte für Mathematik 101, 245–252 (1986). https://doi.org/10.1007/BF01301662
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DOI: https://doi.org/10.1007/BF01301662