Journal of Convex Analysis 19 (2012), No. 3, 685--711
Copyright Heldermann Verlag 2012
M-Structures in Vector-Valued Polynomial Spaces
Dep. de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires -- Argentina
Dep. de Matemática - Pab I, Fac. de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
This paper is concerned with the study of $M$-structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal P_w(^n E, F)$, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal P(^n E, F)$. We show that there is some hope for this to happen only for a finite range of values of $n$. We establish sufficient conditions under which the problem has positive and negative answers and use the obtained results to study the particular cases when $E=\ell_p$ and $F=\ell_q$ or $F$ is a Lorentz sequence space $d(w,q)$. We extend to our setting the notion of property $(M)$ introduced by Kalton which allows us to lift $M$-structures from the linear to the vector-valued polynomial context. Also, when $\mathcal P_w(^n E, F)$ is an $M$-ideal in $\mathcal P(^n E, F)$ we prove a Bishop-Phelps type result for vector-valued polynomials and relate norm-attaining polynomials with farthest points and remotal sets.
Keywords: M-ideals, homogeneous polynomials, weakly continuous polynomials on bounded sets
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