On the spectrum of a nonlinear operator associated with calculation of the norm of a linear vector-functional

An explicit formula is presented for the norm if $1\le p\le\infty$ and for the quasi-norm if $0<p<1$ of a linear vector-functional $L\colon H\to l_p$ on a Hilbert space $H$ and the set of all extremal elements is described. All eigenvalues and eigenvectors of a nonlinear homogeneous operator entering the corresponding Euler's equation, are written out explicitly.