A random multivalued uniform boundedness principle (Q2490463)

The theory of random single-valued linear operators evolved in the 1950s and the theory of multivalued linear operators is even older. The authors present this paper `to begin the process of merging these two areas of research'. Here, multivalued map between two topological vector spaces means that \(T(x)\) is nonempty and closed for each \(x\) in the domain. A random multivalued linear operator is in this paper a multivalued linear operator defined on a Banach space \(X\) and with values in \(L_0 (\Omega,Y)\), the randomization of the Banach space \(Y\). The topology in \(L_0 (\Omega,Y)\) is that of convergence in probability. More concretely, `to begin the process' in this paper means to prove a Uniform Boundedness Principle for random multivalued linear operators. The first step in `merging these two areas' is to have a theory of multivalued continuous linear operators from Banach spaces to classes of topological spaces that are broad enough to contain \(L_0 (\Omega,Y)\), a non-locally convex, non-locally bounded \(F\)-space. Such a theory is developed in Section 2 of the paper, where a Closed Graph theorem, Open Mapping theorem and Uniform Boundedness principle is proved using Baire Category arguments. Let \(\Delta\) denote Hausdorff distance and, for a multivalued linear operator which is continuous with positive probability, let \[ \alpha(T)=\sup\{P(\Omega_0):T| \Omega_0\:\text{is stochastically continuous}\}. \] The main theorem now reads: Let \((T_i)_{i\in I}\) be a family of multivalued random linear operators that are continuous with positive probability and pointwise bounded with probability at least \(\delta>0\) (that is, for every \(x\) in the domain there is an \(M_x>0\) such that \(P[\Delta(T_i(x),T_i (0))\leq M_x]\geq\delta\), \(i\in I\)). Then there exists \(M>0\) such that \[ P[\Delta(T_i(x),T_i (0))\leq M\| x\| ]\geq (2\delta-1)-(1-\alpha(T_i)),\;x\in X,\;i\in I. \]