THE NUMERICAL RADIUS POINTS OF L(2 l(∞,θ)2 : l(∞,θ)2)

For n >= 2 and a Banach space E we let Pi(E)={[x(& lowast;),x(1),& mldr;,x(n)]:x & lowast;(xj)=parallel to x(& lowast;)parallel to=parallel to xj parallel to=1 for j=1,& mldr;,n } L(nE:E) denote the space of all continuous n-linear mappings from E to itself. An element [x & lowast;,x1,& mldr;,xn]is an element of Pi(E) is called a numerical radius point of T is an element of L(nE:E) if |x & lowast;(T(x1,& mldr;,xn))|=v(T), where v(T) is the numerical radius of TT. By Nradius(T) we denote the set of all numerical radius points of T. Let 0 <=theta <=pi/2 and & ell;(infinity,theta)2=R-2 with the rotated supremum norm parallel to(x,y)parallel to(infinity,theta)=max{|xcos theta+ysin theta|, |xsin theta-ycos theta|}. In this paper, we show that the numerical radius of T is an element of L(2 & ell;(infinity,theta)2:& ell;(infinity,theta)2) equals to its norm & Vert;T & Vert;. Using this, we classify Nradius(T) for every T is an element of L(2 & ell;(infinity,theta)2:& ell;(infinity,theta)2) in connection with the norming points of the bilinear mapping associated with TT. Let NA(L(nE:E))={T is an element of L(nE:E):T is norm attaining} and NRA(L(nE:E))={T is an element of L(nE:E):T is numerical radius attaining} We also show that NNA(L(2 & ell;(infinity,theta)2:& ell;(infinity,theta)2))=NRA(L(2 & ell;(infinity,theta)2:& ell;(infinity,theta)2)), which generalizes some results in [12].