Álgebra lineal para Administración y Dirección de Empresas by Emilio Prieto Sáez, Alberto A. Álvarez López

By Emilio Prieto Sáez, Alberto A. Álvarez López

En los capítulos que comprende este texto se exponen los instrumentos matemáticos básicos del Álgebra Lineal, así como una introducción a las sucesiones de números reales. Incluye un tomo con problemas resueltos.

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Suppose that B has an approximute urzit. For each w E Y 4 3 . 9 0 (e being the unit of G ) let p ( w ) be the multiplier of Be obtained by restricting w t o E,. Then p : We(B)+ W ( E , ) is a *-isomorphism (onto all of %‘-(Be)). Proof. It is obvious that p is a *-homomorphism. The preceding proposition states that it is one-to-one and onto W(B,J. 9. 14). A multiplier u of W will be called unitary if u*u = uu* = 4 and llullo I 1. (a) is a group under the multiplication multipliers of 99 by @(a).

7) To do this, we first notice that, for any bounded linear endomorphism F of A, IlFll = sup{IIF(a)bll:a, b E A, llall I1, llbll 5 1 ) = sup{llbF(a)ll:a,hE A, (8) Ilall 5 1, llhll 5 1). Indeed: Clearly IlFll rnajorizes the two suprema in (8). Now,given E > 0, choose a so that llall = 1 and IIF(a)(l > IlFll - E ; and put b = IIF(u)II-~F(u)*. Then llhll = 1 and IIF(a)bII = IIF(a)JJ> - E. Thus the first supremum in (8) equals (JFII. Similarly the second supremum equals 11Fl1. So (8) is proved. If u = ( A , p ) E %‘-(A),it follows from (8) that 11/41 = suP{IlmbII: Ilall, llbll 5 whence = {Ilal(h)ll: IblL llbll 5 1) = 11~11, I> 780 VIII.

A multiplier u of W will be called unitary if u*u = uu* = 4 and llullo I 1. (a) is a group under the multiplication multipliers of 99 by @(a). operation of W(B), the inverse in @(B) being involution. The map, zO:%(A3) --t G sending each unitary multiplier into its order is a group homomorphism. # of order x-that is, if the group homomorphism no of the last paragraph is onto G. Definition. Let u be a unitary multiplier of A3 of order x. The left action of u is norm-decreasing and its inverse is the norm-decreasing left action of u*.

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