Supporting vectors of continuous linear operators

Abstract

The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between the supporting vectors of a real matrix $A$ and the Tikhonov regularization ${min\hspace{0.17em}}_{x\in {\mathbb{R}}^n}\Vert Ax-b\Vert +\alpha \Vert x\Vert$ reaching the conclusion that, by an appropriate choice of $b$ and $\alpha$, the supporting vectors of $A$ can be obtained via solving the Tikhonov regularization ${min\hspace{0.17em}}_{x\in {\mathbb{R}}^n}\Vert Ax-b\Vert +\alpha \Vert x\Vert$.