M-Structures in Vector-Valued Polynomial Spaces

This paper is concerned with the study of

-structures in spaces of polynomials. More precisely, we discuss for

E E

and

F F

Banach spaces, whether the class of weakly continuous on bounded sets

n n

-homogeneous polynomials,

\mathcal P_w(^n E, F) P w ( n E , F )

, is an

M M

-ideal in the space of continuous

n n

-homogeneous polynomials

\mathcal P(^n E, F) P ( n E , F )

. We show that there is some hope for this to happen only for a finite range of values of

n 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 E = ℓ p

and

F=\ell_q F = ℓ q

or

F F

is a Lorentz sequence space

d(w,q) d ( w , q )

. We extend to our setting the notion of property

(M) ( M )

introduced by Kalton which allows us to lift

M M

-structures from the linear to the vector-valued polynomial context. Also, when

\mathcal P_w(^n E, F) P w ( n E , F )

is an

M M

-ideal in

\mathcal P(^n E, F) 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.